Introduction

This is a list of polytempo music compositions, and books or articles that
discuss polytempo music. The list is limited to those pieces in which the
composer has notated the music by using two or more different and simultaneous
tempo markings, and consequently, the duration of a given type of note
(e.g. a quarter note) is not the same in all parts at some point in time.
Here, "tempo marking" means a metronome marking or a tempo term (e.g. allegro
or accelerando).

Pieces for which more than one tempo might be perceived but for which
all parts are written against a common tempo, no tempo markings are specified,
or performers are instructed to deviate freely from specified tempi (e.g.
as in rubato or senza tempo), have been excluded. More specifically:

Pieces where multi-rate effects are notated with polymeters against a global
tempo, have been excluded. For example, in the final section of George
Crumb's Black Angels titled Threnody III: Night of the Electric
Insects, there is a duration of 36 quarter notes at a tempo of MM quarter
note = 60 that is partitioned with six different meters simultaneously
and where at least three different rates of speed may be perceived (i.e.
half-note rate, dotted-quarter rate, and dotted-half rate). And, later
in this section, a metronome marking of MM dotted-quarter note = 36 in
one part is written against other parts that have a metronome marking of
MM quarter note = 54. Since the duration of a quarter note is the same
in all parts, at all points in time, Crumb's piece has been excluded. Similarly,
in measures 93 through 103 of Elliott Carter's Double Concerto for Harpsichord
and Piano with Two Chamber Orchestras, 6/8 measures at a tempo of MM
dotted-quarter note = 70 occur simultaneously with 3/4 measures at MM quarter
note = 105. A similar polymetric structure occurs in measures 619 through
689. Carter's piece has been excluded because the duration of a quarter
note is the same in all parts, at all points in time.

Pieces have been excluded for which multi-rate effects are written with
proportional notation (also called: spatial notation, time-space notation,
proportionate notation or analog notation) and no tempo markings are given.
For example, Conlon Nancarrow's Study No. 8 for Player Piano has
been excluded. Throughout this piece, accelerandos occur simultaneously
with decelerandos. But, the entire piece is written using a form of proportional
notation whereby staccato tones are indicated with flagged eighth notes,
the duration of sustained tones is indicated with horizontal lines that
are appended after noteheads, and the time at which a given note is to
begin is indicated by its horizontal position within the staff. Similarly,
Nancarrow's Study No. 21 for Player Piano (Canon X) in which one
of two voices gradually accelerates throughout, while the other voice gradually
decelerates, has been excluded. And, Nancarrow's Study No. 41 for Player
Piano has been excluded. This work consists of three sections, the
first two of which are for two voices that have different tempi, and the
third section consists of the first two sections performed simultaneously
on two pianos. Similarly, a 22-second section on page 19 of Krzysztof Penderecki's
Dimensionen
der Zeit und der Stille during which six simultaneous unsynchronized
decelerandos occur, has been excluded because the start times for notes
in this passage have been indicated graphically using proportional notation
and no tempo markings are given. Likewise, Luciano Berio's Tempi Concertati
has been excluded.

Mensuration canons of the Renaissance period in which multiple mensuration
signs are provided on a given staff to indicate that a line is to be performed
at different speeds simultaneously, have been excluded. For example, Johannes
Ockeghem's Missa Prolationum has been excluded.

Pieces that contain the multi-rate effects of augmentation or diminution
where such effects have been notated by uniformly lengthening or shortening
the note durations of a given theme, have been excluded. For example, Bach's
fourteenth canon Canon a 4 per Augmentationem et Diminutionem of
the Goldberg Canons (BWV 1087) has been excluded.

Instances of rubato tempo markings or equivalent directions to freely deviate
from the prevailing tempo have been excluded. For example, cases where
a solo part is marked rubato against an accompaniment part that is to remain
in strict tempo have been excluded. Also, "tempo glissandos" (i.e. written,
controlled or measured accelerandos or decelerandos that are embraced by
a broken slur that ends with an arrowhead) such as those found in Elliott
Carter's String Quartet No. 2 (e.g. see the first measure of the
cello part) have been excluded. In the case of this piece, rubato is implied
by the textual directions which specify that such notation indicates an
accelerando where the first and last note of the passage must be played
with a duration as written and "the intervening notes are to be played
as a continuous accelerando (or in other cases, a ritardando, depending
on the notation), the notation indicating approximately whether the accelerando
or ritardando is regular, or more active at the beginning or end of the
passage." (Note: Carter's String Quartet No. 2
has been included in the list given here for other reasons.)

Instances of the marking "senza tempo" have been excluded. For example,
the long "senza tempo" cadenzas beginning in measures I:19 and IV:8 of
the first and fourth movements of György Ligeti's Chamber Concerto
for 13 Instrumentalists have been excluded. (Note: Ligeti's Chamber
Concerto has been included in the list given here for other reasons.)
In this work, the following instructions are given for these passages in
a footnote:

...After attacking, the cadenzas are played as
fast as possible, independent of the metre and bar lines. The rests
in brackets are not to be taken in consideration when playing; they indicate
the imaginary remaining duration of the bars, the cadenzas being notated
as though they had no duration. When a cadenza is finished, the player
waits for the beginning of the next bar; from that point on, the previously
suspended meter is resumed.

NB. When cadenzas begin simultaneously in two or more
instruments, only the attack is simultaneous; after that, the cadenzas
are individual, each instrument playing its notes independently as fast
as possible, and finishing independently. In this way, little time-shifts
can occur; the notation in the score is only an indication, and the individual
cadenzas may end a bit earlier or later.

Pieces have been excluded for which the score appears to indicate different
simultaneous tempi but it is generally believed that the composer did not
intend this to be the case. For example, this might occur if any of the
following conditions is believed to be true: there was an error on the
part of the composer or copyist; the composer considered two different
tempo terms to be equivalent; the composer used different tempo terms to
suggest various moods (or some other characteristics) at a given tempo;
the composer used tempo terms but not to specify tempo. For example, in
his book The Mass in B Minor: The Great
Catholic Mass (p. 295 n.59), Stauffer mentions that he has found
forty-nine instances in the original performance parts for Bach's Mass
in B Minor where different tempo indications (e.g. adagio and lente)
have been used simultaneously. Since it is generally believed that Bach
did not intend to have these parts played at different tempi, this piece
has been excluded.

Any polytempo piece for which there is no known score has been excluded.
For example, electronic pieces for which there is no score have been excluded.

Works such as Henry Brant's American
Weather and Pierre Boulez's Rituel:
In Memoriam Maderna in which the same tempo marking is specified
for all parts, but the parts are to be performed in an uncoordinated manner
(for example, by using independent conductors), have been included. Also,
works such as Charles Ives' Three Harvest
Home Chorales and Symphony No. 4
(specifically, the fourth movement) where at all points in time, one tempo
marking is specified that applies to all parts, but the duration of a given
type of note is not the same in all parts at some point in time, have been
included.

The limitations or restrictions placed on the list given here are not
intended to be taken as definitive properties of the class of music that
is polytempo. Instead, the imposed limitations merely reflect the interests
of the author and also serve to bound the size of the list.

Detailed information regarding instances of simultaneous different tempi
are described in the annotations. Also, the methods by which (and the degrees
to which) performers are to be coordinated or synchronized are discussed.

Compositions

Applebaum, Mark. 1995. Tlön for 3
Conductors and No Players.

This piece calls for 3 conductors that face the audience and lead imaginary
ensembles.
A graphic file of a portion of the score (p.14: marks R, S and T) is
available at http://www.markapplebaum.com/tlon01.html.
In this excerpt, there are three staves, each of which indicates the meter
and tempo (as well as dynamic markings and expression text) for one of
the conductors. For example, the following measures occur between marks
R and S: the first conductor has two measures of meter 9/8 at a tempo of
MM quarter note = 135 (MM dotted-quarter note = 90); the second conductor
has four measures with meters 1/4, 2/8, 1/4 and 2/8 all at a tempo of MM
quarter note = 60; and the third conductor has two measures of 7/8 at MM
quarter note = 105 (MM dotted-quarter note = 70). For these measures, all
barlines are common and drawn through all parts, except the right barlines
of the two 1/4 measures in the part of the second conductor. A description
of the piece is available at http://www.markapplebaum.com/acoustic.html.

On page 25, the nine instruments are partitioned into three groups
of three. The first group (trombone, bass clarinet, bassoon) plays a 3-measure
phrase with meters 7/8, 3/4 and 5/8 at MM eighth note = about 114. At the
fifth eighth note of this phrase, the second group (viola, cello, double
bass) begins a 4-measure phrase with meters 4/4, 7/8, 4/4 and 7/8 at MM
eighth note = 152. At the fourth eighth note of the third measure of the
first phrase, the third group (flute, clarinet, harp) begins a 3-measure
phrase with meters 7/8, 4/4 and 7/8 at MM eighth note = about 133. None
of the barlines of these simultaneous phrases coincide. Each phrase ends
with a long fermata. This page is quoted in the book Modern
Rhythmic Notation by Gardner Read (p. 170).

Arrigo-Nelson, John 2000. Temporal Quartets.

The eight instruments are organized as a string quartet and a mixed
quartet (oboe, alto flute, vibraphone and trombone). The tempo markings
for the quartets are independent. Over the course of the piece, the tempo
of the string quartet decreases in a step-wise manner through the following
sequence of tempi: MM quarter note = 112, 84, 70, 58 and 48. Concurrently,
the tempo of the mixed quartet increases step-wise through the following
sequence: MM quarter note = 48, 58, 70, 84 and 112. (Note: All tempi are
in the metronome scale, except 70.) The times at which the quartets change
tempo do not coincide.

The performance notes call for two conductors. Usually, the quartets
play in an uncoordinated manner, relative to each other. There are two
times at which the beats of the quartets are to be coincident. During 5
measures in the middle of the piece, both quartets play at the tempo MM
quarter note = 70, with common barlines. The first beat of this section
is coordinated by having the members of the string quartet play unmeasured,
trilled glissandi while the mixed quartet plays two (preparatory) measures
at MM quarter note = 70. At the end of the piece, the quartets play the
final chord simultaneously. One of the conductors provides a cue for this
event. Here, the duration of rests that immediately precede the final chord
in each quartet, is adjusted, as needed.

There are beats or sub-beats that would coincide if the tempo of each
quartet were followed precisely. Dashed vertical lines are used to indicate
these. (Note: Coincident barlines are not drawn through from one quartet
to another.) In some cases, a rhythm line is provided between the quartet
parts to indicate sub-beats associated with tuplets. Such beats and sub-beats
are not required to coincide in performance. In the performance notes,
Arrigo-Nelson writes:

I have placed occasional dotted vertical lines in the
score where certain beats or events line up. These vertical lines, however,
are simply a guide as to where these events occur. My overall concern is
the effect of the two tempi existing simultaneously.

System breaks are placed in the score at barlines of the quartet that
is playing at the faster tempo. The measures of the slower quartet are
broken across systems. In broken measures, the following notational devices
are used: broken beams; arrows pointing to the right that are written on
the middle staff line at the end of one system and at the beginning of
the subsequent system; and broken tuplet brackets with an arrow pointing
right after the tuplet number at the end of a system, and an arrow pointing
left before the tuplet number at the beginning of the subsequent system.

Arrigo-Nelson, John 2002. Solo/Duo for Violin and Piano.

Often in this work, the tempi of the violin and piano are different.
For example, in the opening, there is a ritardando in the piano from MM
quarter note = 112 to MM quarter note = 69, while the violin continues
at MM quarter note = 112. Later, the tempo of the violin is MM quarter
note = 45 while that of the piano is MM quarter note = 120. During another
period, the tempi of the violin and piano are MM quarter note = c. 80 and
MM quarter note = c. 36, respectively.

Throughout, the parts are notated so that the horizontal position of
events indicates the relative times at which events are to occur. Generally,
the instruments are to be loosely coordinated. In the performance notes,
Arrigo-Nelson writes:

In sections of the score where the instruments are playing
in independent tempi, the music of each part is written in correct proportion
to the other. It is not expected that every event in these areas will occur
exactly as written on the page. The events should occur, however, in the
general area in which they are written.

Several events, which are listed in the performance notes, are to be
coordinated between the two parts. This is accomplished by adjusting the
duration (or number of repetitions) of material that precedes the coordinated
events, as needed.

This 11-minute work is an eight-voice canon in which eight musicians
perform four voices along with four voices of tape music that was generated
with a Synclavier Digital Music System.
Each voice enters as an exact melodic and rhythmic imitation but at a tempo
that is different from all other voices. Voices gradually (and independently)
accelerate and decelerate over long periods of time. Four audio beat-cue
tracks are played from tape and distributed to the eight musicians to enable
the players to remain synchronized with each other and with the tape music.
At five points in time, all eight voices momentarily play in unison. Program
notes for this piece may be found at http://www.music.unt.edu/cemi/LApnotes.htm#canadian.
The score is available at the American Music
Center (call number: AMC M1613.3 A936 C2).

The viola opens the first song Black is the Colour... with some
introductory material at MM quarter note = about 72 after which the voice
and harp enter at MM quarter note = 54 while the viola continues at MM
quarter note = 72 "indipendente dal canto" (independent from the singing).
Then the viola plays alone at MM quarter note = about 90. The voice and
harp join in at MM quarter note = 54 and the viola accompanies them at
MM quarter note = 94. Again the viola plays alone at MM quarter note =
72 but then switches to MM quarter note = 92 as the voice and harp enter.
Gradually, over the course of this passage the viola slows down to MM quarter
note = 54, and for a brief period, all play at the same tempo. The song
concludes with seven 3/4 measures for the violin and cello at MM quarter
note = 92. Most of the viola passages in this song are unmeasured.

In the song Motettu de Tristura (for viola, cello, 2 percussionists,
voice, harp and piccolo), in the section that follows mark number 2, the
voice is at MM eighth note = 60 while the piccolo is at MM eighth note
= about 72 "liberamente, senza badare al canto" (freely, without paying
attention to the singing). The following footnote is given: "It is not
necessary to synchronize the parts. At the entrance of ottavino [piccolo]
the conductor should stop giving cues. Viola and Cello should stop playing
last.".

The orchestra is divided into eight groups, each with a percussionist.
The piece consists of a sequence of 15 segments or sections. In the odd-numbered
segments, Group 8 (which has two percussionists) plays at MM half note
= about 50-54 (très lent) while the conductor improvises the cue
time for each of a sequence of chords to be played by the rest of the orchestra.
The even-numbered segments are played asynchronously. In these segments,
each of the groups 1 through 7 is given a phrase to be played at a tempo
of MM eighth note = about 80-92 (moderato). The conductor cues the entry
of each phrase at a time and order as he chooses, with the constraints
that the first phrase that is played must be the longest and that all phrases
must begin and end within the duration of this longest phrase. The percussionist
in each group maintains the beat for his group. All beats are of equal
duration but the times at which beats occur is not necessarily the same
in all groups.

The orchestra is divided into five widely spatially-separated groups:
strings (or clarinet choir), woodwinds, horns, muted brass, percussion.
Each group is positioned in a different part of the hall and is notated
with a distinct tempo, meter and barline scheme. The strings are led by
a principal conductor and there are four section-conductors for the other
groups. An optional voice group may be added with its own conductor. The
principal conductor cues the entrances of the section-conductors. In the
"Explanatory Remarks", Brant writes:

Each section conductor must take his own tempo (specifically
indicated) from the beginning of each entrance, and then proceed, clearly
and steadily, entirely independently of all the other conductors.

Also, he writes:

It is not expected that the relationships between the
different barlines in the score will be carried out precisely as indicated.
What is intended is an approximate relationship, but the approximation
should be as close to the score as possible, although it will be apparent
that a certain small amount of leeway is feasible, in the timing
of entrance cues, and in permitting certain groups to finish their phrases
slightly before or after the points indicated.

This was the first spatial (or antiphonal) work by Brant. He has composed
dozens of pieces of this type. In all cases, the various ensembles are
to be widely separated, and in most cases, a different metronome marking
is specified for each simultaneous musical phrase or fragment. A biography
of Henry Brant may be found at http://www.carlfischer.com/brantbio.html
and a list of his works which contains a subsection titled "Spatial Works"
may be found at http://www.carlfischer.com/brantworks.html.
The scores for most of the works that are discussed in the bibliography
given here are available at The
New York Public Library for the Performing Arts. A complete set of
Brant's manuscripts is contained in the collection of the Paul Sacher Foundation
in Basel, Switzerland.

Brant, Henry 1954. Millenium 2. New York: Carl Fischer.

This work contains early instances of many devices that Brant has used
again in later works. For example, there are looped phrases that are to
begin on cue and then be played in an uncoordinated manner. Upon hearing
the sound of some musical event, such phrases are to be stopped after a
stop mark has been encountered in the score. Often, several phrases of
this type overlap. In some cases, they begin sequentially, one by one,
while in other cases they begin at the same time. Also, there are phrases
that are to be played once (with no repeat), in an uncoordinated manner.
Simultaneously with uncoordinated phrases such as these, there are coordinated
phrases that are conducted. Each phrase is allowed to have a tempo that
is independent of that of any other phrase that is being performed simultaneously.

In the notes, Brant writes the following:

The horns, baritone horns and tuba, seated on the stage,
follow the conductor precisely, in the normal way, from beginning to end.
The timpani, gongs and thundersheet likewise; and from (M) to the end,
the bell instruments also.

However, trumpets 1 to 9, and trombones 1 to 9, from the
beginning to (V), all work on a different co-ordination plan. Each of these
players enters on a direct, individual cue from the conductor, after
which he ignores the conductor and all other players, and proceeds for
himself, making no effort to maintain ensemble. The individual player,
under these conditions, should nevertheless make every effort to perform
in as steady and accurate a manner as possible. In most cases these entrances
are divided up into phrases which are indicated by the termination sign
[right bracket symbol]. After several phrases a repeat sign appears. The
player is instructed to play through each such sequence, repeating if necessary
in order to keep playing until a particular designated point, an obvious
place such as a loud percussion entrance. When he hears this "cue", he
continues
playing until the end of his current phrase [right bracket symbol]
and there makes a clean and definite exit. No "fade-out" or sudden "cut-off
on cue."

At (S) and elsewhere a simpler device is used, there are
no indicated phrase divisions or repeats; the player merely enters on cue,
plays everything in the sequence once and then stops.

At (G) the 9 trumpets and 9 trombones enter precisely
together, with the conductor, and then immediately go their separate
ways. The trumpets alone at (Q), and the trombones at (S) have this same
briefly simultaneous, and then promptly scattered, procedure. The bell
instruments have this same kind of thing at letter (B).

In other sections, especially in the long (M) sequence,
the conductor has the task of maintaining precise ensemble for the horns,
tubas and percussion, and at the same time, giving direct individual cues
to particular trumpet-trombone pairs or sections. This is workable because
these "free" entrances always occur in relation to specific down-beats
of the particular tempo which the conductor happens to be maintaining at
these points.

From (V) to the end, the tempo system becomes normal,
the entire group moving together in the same metre and tempo.

Here, the right bracket symbol that Brant uses as a stop mark is similar
to the mark that has been used by others to indicate the end of a passage
that is to be prominent as in Schoenberg's Hauptstimme and Nebenstimme.

This work is for strings, woodwind trio (flute, clarinet, bassoon),
solo flute, solo horn, solo oboe, glockenspiel and one conductor. There
is also a version for wind ensemble. The conductor leads a unison part
for strings (violins, violas and cellos) that consists of a sequence of
five sections that have the following tempi: MM quarter note = about 104,
about 60, about 104, about 60, and about 104. Sometimes, the woodwind trio
is synchronized with the strings, and when used, the basses are synchronized
with the strings, though not in unison. Other parts which have a different
tempo are cued by the conductor and then continue unconducted. The following
tempi are used for such parts: MM quarter note = about 84 (oboe), MM half
note = 72 or slower (horn), MM quarter note = about 80 (flute), MM quarter
note = 72 or slower (woodwind trio, when not synchronized with strings).

Phrases are repeated in order to control the total duration of a given
passage and to ensure that a passage ends before or after the start or
end of some other passage. To achieve this form of control in the string
part, which has no repeats, the conductor may continuously adjust the tempo
as needed. In repeated phrases, stop marks in the form of breath marks
or bold, nearly-horizontal, upward-sloping slash symbols are placed over
rests. In such phrases, when the conductor issues a stop cue, the player
continues to the next stop mark and stops, or jumps to a repeat ending,
depending on the case.

This work calls for two spatially separated groups of instruments.
Throughout, Group I plays in the meter 4/4 at MM quarter note = 94-100,
and Group II plays in the meter 3/4 at MM quarter note = 72-80. All parts
share a common barline throughout. That is, the duration of each 4/4 measure
for Group I equals that of a simultaneous 3/4 measure for Group II. In
a section titled "Rhythmic Co-ordination" of the performance notes, Brant
writes:

Each group has its own conductor, and the seatings are
so arranged that each musician sees only his own conductor. Group I plays
in 4/4 meter, Group II in 3/4, throughout. One measure of 4/4 is equal
to one measure of 3/4; the first beat of each 4/4 measure must coincide
with the first beat of the corresponding 3/4 measure, and the resulting
effect will be that of "3 against 4" throughout, whenever both groups
are heard simultaneously. In order to achieve this 3:4 ratio as easily
and securely as possible, it is essential that the two conductors face
each other directly.

The entire piece is in a single overall tempo which does
not change, although a minimal flexibility is expected; it should be kept
even and steady. This tempo is MM quarter note = 94-100 for the 4/4 group,
which corresponds to MM quarter note = 72-80 for the 3/4 group.

Depending on the acoustic properties of the hall and the
distance between the two groups, the conductors may wish to devise further
aids to co-ordination, such as a "click-track" or a flashing light at the
beginning of each bar (24 times per minute); or instead, a cuing system
so that either conductor may stop beating during extended rests, and be
cued in by his colleague one bar before his next entrance. These are suggestions
only. Any accurate solution of the 3:4 ratio is satisfactory.

Brant, Henry 1973. Divinity: Dialogues in the Form of Secret Portraits
for Harpsichord and Brass Quintet, Spatially Separated. New York: Carl
Fischer.

No conductor is required. Each player reads from the full score. In
the score, entry times are indicated relative to the music of instruments
that are already playing. In the "Directions for Performance", Brant discusses
the terms co-ordinated and un-co-ordinated:

The term CO-ORDINATED means that all the players have
the same metre, tempo and barlines; and that they maintain exact rhythmic
ensemble together in the usual way. UN-CO-ORDINATED, or NON-CO-ORDINATED,
as used here, means that each instrument, or group of instruments, has
an individual tempo and metre. Each such tempo should be maintained steadily
and accurately, but the players should make no attempt to "keep together".
Each tempo ignores the others.

Uncoordinated playing is specified when there are simultaneous different
tempi. And, it is used in some cases when all members of an ensemble have
the same tempo marking (which is individually specified for each instrument
in the ensemble).

Each page of the score is divided vertically into sections by bold horizontal
lines, where each section contains a set of independent systems that are
to occur simultaneously.

As in all of Brant's works (that have been seen by this author), traditional
metronome numbers are used for all tempi. For example, the following numbers
are used here: 63, 66, 69, 76, 80, 84, 92, 116, 132, 144 and 176.

In the introductory notes, Brant writes the following regarding the
mode of performance and the structure of the score:

This piece should, if at all possible, be performed in
a dark hall, with all instruments invisible to the audience. It may thus
happen that the players will be unable to see each other, but this will
not create a problem, since no exact rhythmic co-ordination between
the players is intended, and no visual contact between them necessary.
All can be accomplished via sound cues, but this obviously makes it essential
for the players to be able to hear each other.

There can be no "score" of this work, in the usual meaning
of the term, because the five parts are completely independent of each
other in tempo and rhythmic co-ordination, so that no attempt to represent
vertically what takes place when the instruments all play, can be accurate.
The present form of the score gives the entire musical material of the
piece fully written out, the musical "events" indicated in the sequence
in which they occur.

The piece consists of a sequence of 7 sections. A structure that consists
of a sequence of progressively faster sections is used thematically throughout
the work. The tempo markings for the first three subsections of Section
1 are: "slow", "moderate" and "faster still". In each of Sections 2 through
6, one of the winds plays a "solo" part. The tempo of each solo is faster
than (or the same as) that of the previous solo, and is as follows:

In Sections 2 through 6, the soloist is accompanied by the other four
instruments which enter sequentially and play looped phrases that are in
a tempo that is different from that of the soloist. The accompaniment passages
begin at one of following types of points: a 1-measure rest within the
solo part, after a "jazz-lick" in the previous solo part, shortly after
another accompaniment begins, or (more rarely) during a sustained solo
note. The accompanists complete their current figure upon hearing the "jazz-lick"
that ends each solo. In each section, all accompaniment passages are based
upon a particular type of figure (e.g. a trill, sustained note, grace note
or repeated notes). The figures are separated by pauses of irregular or
varying length.

Repeated phrases with breath marks or slash symbols are used as in
Brant's On the Nature of Things.
Here, fermatas are also used as stop marks. Start and stop cues are issued
by the conductor (who does not beat time at all during the piece) or by
easily recognizable phrases such as a loud oboe chord or a piccolo motif.
Time intervals during which the conductor is to cue the entry of an instrument
are indicated with text above or below the staff of an instrument that
is playing at the time of entry (e.g. "CUE HARP").

A linear polytempo texture consisting of the sequential accumulation
of different repeated phrases, occurs throughout the piece. For example,
in the second segment, a 7-measure phrase is played repeatedly on the bassoon
at MM quarter note = 69. During the fourth measure of this phrase, the
conductor cues the harp to begin playing a 4-measure phrase repeatedly
at MM quarter note = 66. During the second or third measure of the harp,
the conductor cues a trombone to begin playing an 11-measure phrase repeatedly
at MM half note = 76. Then, the conductor cues a short piccolo phrase which
is used throughout the piece as a cueing device. Upon hearing the piccolo,
the bassoon, harp and trombone complete their phrases through to their
repeat barlines.

Vertical (or harmonic) textures occur in which all instruments play
at the same tempo but in an uncoordinated manner. For example, upon hearing
the piccolo (at the end of the previous example given here), the viola,
cello, double bass and piano begin playing different 2-measure phrases
repeatedly at the tempo MM quarter note = 66. Here, the players are instructed
to "begin as soon as possible but without attempting to start at the same
exact moment".

A canon-like texture occurs in the violins. Here, a 20-measure phrase
is played repeatedly and simultaneously by 10 violins at MM half note =
63, where each violin begins playing at a different odd-numbered measure
within the phrase. The violins are instructed to begin "'at once' but not
attempting an exact simultaneous attack".

At various points throughout the piece, these textures (linear, vertical
and canon) are performed simultaneously.

In this piece there are two groups. Group I consists of a full choir
with a trombone and chimes. Group II consists of eight solo voices, a trumpet
and a glockenspiel. Group I is lead by the conductor and performs throughout
as a coordinated ensemble at the tempo MM half note = 96. At four points
in the piece (marks 5, 12a, 20a and 28a), the conductor cues the entry
of the trumpet. After the trumpet plays a few measures, the entrance of
the eight solo voices is cued by the glockenspiel. The eight solo voices
attack their opening note together and are instructed to "then proceed
independently, making no effort to keep together". The conductor does not
beat time for the trumpet, glockenspiel or solo voices. The tempo marking
for all the parts in Group II is the same as that of Group I (i.e. MM half
note = 96), but Group II is not coordinated with Group I and in general,
members of Group II are not coordinated among themselves.

Uncoordinated repeated phrases occur at a tempo that is different from
that of music with which they are simultaneous. Also, there are phrases
that are uncoordinated after the first note where each instrument of the
group has the same tempo marking.

Brant, Henry 1977. Cerberus: after "A Report to an Academy" (Kafka)
for Piccolo and Double Bass. New York: Carl Fischer.

The bass plays throughout and serves to cue the entry of the piccolo.
During the course of the piece, the tempo of the bass is changed twice.
It begins at MM quarter note = 63, changes to MM quarter note = about 152,
and then returns to MM quarter note = 63. Throughout, the piccolo plays
various phrases each of which ends with a final barline. Each piccolo phrase
begins at a rest in the bass part, usually at a down beat in the bass.
When the tempo of the bass is MM quarter note = 63, the piccolo plays at
MM quarter note = 160. When the bass changes to MM quarter note = about
152, the piccolo changes to MM quarter note = 92. Whenever the piccolo
enters, the bass rests a given number of beats in its own tempo, and then
continues playing. Generally, the bass begins playing again before the
piccolo has completed its phrase, and therefore at these times, the instruments
play simultaneously at a different tempo.

One conductor leads two separated groups (or an assistant conductor
may be used to lead one group, in which case the assistant duplicates the
principal conductor's beats exactly). One group is located on stage while
the other is at the back of the hall. The tempo of Group I is MM dotted-quarter
note = 84 and the meter is 6/8. The tempo of Group II is MM quarter note
= 84 and the meter is 2/4. There are no meter or tempo changes. The barlines
of the groups are horizontally aligned and the ensembles are to be coordinated
in the normal way. The piece consists of a sequence of 34 subsections that
are divided into 7 sections.

The eight widely separated instruments are: piccolo, clarinet, English
horn, trumpet, trombone, {timpani, xylophone or chimes}, {glockenspiel
or vibraphone} and harp. Throughout various sections, these instruments
play repeated phrases at a tempo that is different from that of other instruments.
The conductor cues their entry after which each continues to play without
direction, uncoordinated with other music. For example, at mark 4 the conductor
cues the entry of the harp, timpani and vibraphone in relation to the bassoons.
Here the bassoons have a tempo of MM half note = 56. The three entering
instruments (whose entry is staggered) have a tempo marking of "Slowly,
in irregular, free rhythm". At another point, the five wind and brass solo
instruments are cued by the conductor, one at a time, in relation to the
bassoons. Here the bassoons play at MM half note = 76 and the five entering
instruments have a tempo marking of "Moderately fast" or "Fast 16ths".

This work calls for an orchestra that is to be divided into three mixed,
spatially separated groups (A, B and C). Each group has its own conductor
and plays in the same tempo and meter throughout as follows: Group A: meter
4/4 at MM quarter note = 84, Group B: alternating between meters 2/4 and
3/4 at MM quarter note = 100, and Group C: meter 3/4 at MM quarter note
= 60. An exception to this occurs in Sections 4, 7, 14 and 16 where Group
A may be slowed to MM quarter note = 76 if its normal tempo (MM quarter
note = 84) is uncomfortable.

The work consists of a sequence of phrases each of which is played by
a given group. The entry of each (child) phrase is cued by a conductor
in response to a percussive signal that is heard in a (parent) phrase that
is being played by some other group. For any sound cues that are found
to be inaudible, the conductor of the parent phrase may give a visual cue
to the conductor of a child phrase that is about to enter.

In the "Notes on Performance", Brant writes:

Each conductor should make every effort to retain his
tempo and to keep it steady, but it is not intended that an exact correspondence
of the barlines of the 3 groups should be maintained. The result will
be that the groups, although playing in accurate ensemble separately, will
not be in precise rhythmic synchronization all together.

Accordingly, each conductor, upon hearing the percussion
signal and seeing that his musicians are ready to play, may begin immediately
without trying to make his first attack coincide with the downbeat of another
group as indicated in the score. Once begun, each conductor should lead
each passage through steadily to its end without being influenced by what
the other groups may be playing.

There are, however, a few places where the conductors
should make every effort to make entrances coincide as far as possible
with the downbeat of another group as indicated in the score. These are
at 1a and 1b, and at 17a and 17b. See also the note in Group C at 16.

The following forces are used: an orchestra, a "wall brass" (i.e. 3
trumpets and 3 trombones in single file against a side wall), two percussion
ensembles, a jazz orchestra, two soprano soloists, a chorus with saxophones,
a chorus with piccolos, a Javanese gamelan orchestra, a West African drumming
ensemble, and a South Indian trio. The principal conductor leads the orchestra,
wall brass, the two percussion ensembles and sometimes the saxophones and
piccolos that are associated with the choral groups. Each choral group
has its own conductor. The jazz orchestra is lead by its drummer.

The principal conductor cues the entry of the solo sopranos, the South
Indian trio, the gamelan orchestra, the African drummers, and the jazz
drummer. In Appendix II (p.53) titled "Notes on Cues and Signals", 14 cue
gestures are suggested, each of which may be used to cue a particular ensemble
or individual. These gestures have grown out of Brant's experience with
spatial music and background music for live radio. It is suggested that
a cue gesture be issued with the left hand to gain the attention of the
entering player(s) and that this be followed by an up and down-beat with
the (right hand) baton alone. For example, for "cueing a large group out
in the hall, at the conductor's right", the following gesture is suggested:
"left arm curved over the head, with index finger pointing to the right".

The piece consists of a sequence of 17 sections. Different forms and
combinations of ensembles are used for each section. Various polytempo
structures are used, some of which occur in earlier works by Brant. For
example, in Section 5, loops of repeated phrases with the indication "fast
32nds, irregular 1/4 rests" are played by the percussion groups while the
orchestra plays at MM quarter note = 72. Later, in Section 11, the strings
begin looped phrases at MM quarter note = about 104. Then, on cue, the
flutes and clarinets begin (together) two different staggered canons (like
that of the violins in Brant's Curriculum)
at MM quarter note = about 116. A chord which is repeated three times by
the oboes, bassoons and horns is then used to cue, in sequence: a looped
phrase in the solo sopranos (with no tempo marking), a looped phrase for
Chorus II at MM quarter note = 80, and a looped phrase in Chorus I at MM
quarter note = 56. All parts (except the cue chords) continue to play through
to the end of the section and then stop after the West African drumming
ensemble enters to start Section 12.

Throughout the full score, musical material is abbreviated. Some phrases
are abbreviated symbolically as "X", "Y" or "Z" (e.g. see: p.33) or with
numbers (e.g. see numbered songs: p.88). The complete material for such
phrases is contained in Appendix I.

In Section 16, the percussionists are instructed to look up upon encountering
a fermata in their part to see if the conductor is issuing a (sustained)
stop cue. Here, the text "look up" is written above each fermata (see:
Appendix I: pp.39-42).

There are three separate scores: a piano part score; a score for a
woodwind group, string group and lights (for projected slides of photographs
of paintings by Leslie Parke); and a score for band (trumpet, trombone,
horn, alto/baritone saxophone, drummer and sopranino voice). The woodwind
and string groups are lead by a principal conductor and the band is lead
by an assistant conductor who may be the vocalist.

Like Brant's Meteor Farm, the
piece consists of a sequence of 17 sections. The band plays alone in the
odd-numbered sections through section 15. The piano, woodwinds and strings
play the even-numbered sections as well as the last section (number 17).

The band and piano scores contains no cue information from other parts.
A one or two-measure ossia of the beginning of the piano part is given
at the beginning of each even section in the woodwind/string score, and
the entire piano part for section 17 is notated in this score with a meter
and tempo that matches that of the winds and strings. There are no meter
or tempo changes within a section in any part, except in the piano part
where changes of meter (but not tempo) occur often within a section.

In the even sections, the winds and strings play synchronously through
material that is repeated a number of times (though either the winds or
strings are omitted from some repeats). Against this background, the piano
plays in an independent meter and tempo. The tempo combinations for the
17 sections are as follows (p = piano, w/s = winds/strings):

The string section is lead by the principal conductor who beats time
for this group. A group of high woodwinds (3 piccolos, 3 oboes, and 3 clarinets)
which are to be positioned in the topmost balcony at the back of the hall,
are coordinated by an assistant conductor. Generally, the other players
do not follow a conductor's beat. Instead, their entry is cued by the principal
conductor or by the sound of some musical event (without a visual cue),
and thereafter they play in an uncoordinated manner. The time at which
conductor cues are to be given is not indicated in the score by vertical
lines that relate one part to another, though for some parts, textual notes
instruct the conductor to be guided by the relative horizontal position
of an entering part and other parts that are playing, in which case, such
entries are to be made "approximately at the point indicated". In many
cases, the members of a group that has been cued are to enter in such a
way that their first notes are not necessarily coordinated.

Simultaneous different tempi occur throughout the piece. For example,
the first section contains the following sequence of events:

On a cue from the principal conductor, the timpani begins a 10-measure
phrase at MM quarter note = 76-80 (meter 3/4) and then continues independently,
without being lead by the conductor. The phrase contains one fermata.

At some time during the first few measures of the timpani part, the low
woodwind group (bass clarinet, 2 bassoons, and contrabassoon) and tuba
are cued by the principal conductor to begin playing looped phrases at
MM quarter note = 92 (meter 4/4). This group plays their first notes together,
and then they play independently of each other thereafter. Each instrument
has a phrase of a different length (bass clarinet and contrabassoon: 17
measures; bassoons: 18 measures; tuba: 15 measures) and each phrase has
two to four fermatas which serve as stop marks.

Near the end of the timpani phrase, the conductor cues the tom-tom and
bass drum to begin playing a 10-measure phrase at MM quarter note = 76-80
(meter 3/4). The two instruments are to be coordinated among themselves
but are not lead by the conductor. The phrase contains one fermata.

During the first few measures of the tom-tom and bass drum phrase, the
conductor cues the five horns to begin playing material that is structurally
similar to that which the low woodwinds and tuba have been playing, and
like this other group, the horns play at MM quarter note = 92 (meter 4/4).

Before the end of the tom-tom/bass drum phrase, the conductor cues the
entry of the strings which play a 22-measure phrase at MM half note = 60
("or a little slower - but not faster") (meter 2/2). The conductor beats
time for the strings. The string phrase has no fermatas.

At the 7th measure of the string phrase, the conductor cues the timpani
to begin a 4-measure phrase at MM half note = 76-80 (meter 3/4).

At the 11th measure of the string phrase, the conductor cues the tom-tom
and bass drum to begin a 4-measure phrase at MM half note = 76-80 (meter
3/4).

At the 17th measure of the string phrase, the conductor cues the timpani
to begin a 4-measure phrase at MM half note = 76-80 (meter 3/4).

The first section of the piece ends with the strings holding a fermata
at their 22nd measure, and the low winds, tuba, and horns all continuing
their looped phrases. During this time, the conductor cues the entry of
the "Bell Trio" (i.e. glockenspiel, vibraphone and chimes) to begin playing
the second section of the piece at MM quarter note = 76. Upon hearing the
bell trio, each of the instruments that is playing a looped phrase stops
after playing through to the next fermata in their own part.

The prime numbers 7 and 17 appear often in the works of Brant. For
example, many of his works consist of a sequence of 7 or 17 sections. This
piece has 7 sections, and in the first section, cued phrases are to begin
at the 7th and 17th measures of the string phrase. Also, sometimes Brant
uses the double of a prime number. For example, some works have 34=2*17
sections. The string phrase in the first section of this piece is 22=2*11
measures long.

In the section titled "Co-ordination and Ensemble" of the "Performance
Notes", Brant writes the following regarding the term "unco-ordinated":

The term "unco-ordinated" is not intended to suggest rhythmically
casual or imprecise playing. Accurate rhythm is desired with careful attention
to all indications of tempo, articulation, and dynamics. However, in playing
an "unco-ordinated" passage, the performer is asked to maintain an individual
correctness in execution, without attempting to "match" or "fit in", and
without following a beat.

The 80 trombones are partitioned into eight spatially separated groups
(A through H) of ten each. Some uncoordinated textures occur in this work.
For example, in Section 1c, which is a 1-measure section for Groups A through
D, the instruction "Unco-ordinated (after first note)" is given. Here,
ten phrases are played simultaneously with four trombones assigned to each
phrase. The phrases contain a mix of straight eighth notes as well as eighth-note
tuplets with the ratios 5:4, 3:2 and 7:4. This is followed by a similar
uncoordinated 1-measure Section 1d for Groups E through H. Later, in the
1-measure Section 7a, 20 different repeated phrases are played simultaneously
by all 80 trombones, with four trombones per phrase, and all play in an
uncoordinated manner (after the first note) at MM quarter note = 80. A
similar texture is used in Section 9a at a tempo of MM quarter note = 176.
Outside of these uncoordinated measures, all parts share the same tempo
marking and play in a coordinated manner throughout.

There are many measures in this piece for which the duration of a given
type of note is not the same in all parts. Some instances where this occurs
in the first movement are as follows: In measure 30, the tempo of the viola
is MM dotted half note = 60 while the other instruments play at MM quarter
note = 120; In measure 37, the cello plays at MM half note = 90 while the
other instruments play at MM 5 eighth notes = 90; In measure 45, violin
I plays at MM quarter note = 150 while the others play at MM 5 eighth notes
= 60; In measure 46 violin I plays at MM 5 eighth notes = 120 while the
others play at MM half note = 60. Similar instances also occur in the second
movement. For example, in measures 85 through 125 of the second movement,
the violins play at MM eighth note = 108 while the viola and cello play
at MM eighth note = 72. In all the measures cited here, all parts share
a common right barline but not a common meter. For example, in measure
37 of the first movement, the meter of the cello is 3/2 while the meter
of the other instruments is 15/8. In all these cases, a tempo marking is
given to explicitly indicate the difference in tempo that results from
such a polymetric configuration.

In the second movement, the part of the second violin has been approximated
and written in the meter and tempo of all other parts (i.e. 5/4 at the
tempo MM quarter note = 175). A rhythmic cue line and explanatory footnote
are given to indicate that this part is to be played in the meter 4/4 at
the tempo MM quarter note = 140.

The instruments are partitioned into two groups of two: Duo I (violin
I and cello) and Duo II (violin II and viola). Generally, the piece is
notated in such a way that the duration of a given type of note is the
same in both parts at all times. However, an exception occurs in measures
26 through 28 where Duo I plays a 4/4 measure and 2/4 measure at MM half
note = 63 followed by a 3/4 measure at MM quarter note = 108, and Duo II
plays a 2/4 measure at MM quarter note = 189 followed by a 3/4 measure
and 4/4 measure at MM quarter note = 108. The left barline of measure 26
and the right barline of measure 28 coincide. This is not an approximation
(because 6/126 = 2/189 + 4/108).

The piece calls for two conductors and for most of the time these conductors
establish two simultaneous independent tempi. The orchestra is not divided
into two fixed groups. Instead, any given instrument might be conducted
by a different conductor at different points in the piece. In the "Conductors"
section of the "Program Notes", Colgrass writes:

This work requires 2 conductors because of multi-tempo
musics. The orchestral musicians have either a I or a II shown in their
parts next to each entrance so that they may simply look to the appropriate
conductor for each cue. Although the music as written should be played
as accurately as possible, I do not expect the 2 conductors to be necessarily
in exact coincidence, bar for bar, at all times. Music-making is more important
to me here than computer precision. There will usually be a fermata of
other obvious "catch-up" point for the conductors to become re-oriented.

Constant, Marius 1968. Winds. Paris: Editions Salabert.

On pages 18 and 19 of the 29-page score, the following tempi occur
simultaneously: MM quarter note = 69 in 2/4 (horns), MM quarter note =
126 in 3/4 (trumpets), MM dotted-quarter note = 69 in 6/8 (trombones),
and MM quarter note = 63 in 4/4 (tuba). A rhythmic schematic of this is
given in the book Modern Rhythmic Notation
by Gardner Read (p. 172).

This work calls for four conductors: two primary and two secondary.
The performers are spatially separated. Some groups are positioned in the
audience. In the "Program Note", Crumb writes:

Star-Child is continuous, despite sectional divisions.
The germinal idea, "Music of the Spheres" (strings, pianissimo),
moves throughout the work in a circular music over which the human drama
is enacted. This idea consists of a continuum of chords built upon the
interval of a perfect fifth. Over these slow-moving strains of "suspended"
music I have superimposed (in the manner of Charles Ives!) a sequence of
boldly contrasting musics. The necessity for four conductors arose from
the fact that each music has its own tempo and metrics (metrics tend to
be odd-numbered: the opening string music is in 11/4 time, the entire Apocalyptica
in 5/16, and there are other sections based on sevens and threes). The
four conductors do not synchronize and therefore all sense of vertical
alignment is erased. I had even imagined that the "visual counterpoint"
of the four-fold conducting would produce a choreography of its own!

The "circular" music to which Crumb refers is music that is notated
on two curved staves that together form a circle. Against this background
of looped phrases (that are structurally similar to the looped phrases
found in the works of Henry Brant) music of a different tempo is played.
The looped phrases are conducted by one of the conductors II, III or IV,
or in one case, performed by one instrument (i.e. the organ) without a
conductor. Such phrases are repeated until a stop cue is issued. At a stop
cue, players are instructed to complete the circle and then, depending
upon the particular phrase, either stop playing, or proceed to a repeat
ending.

This work is for three spatially separated choirs of four soloists
each (SATB). It consists of five sections titled "Kyrie", "Gloria", "Sanctus",
"Benedictus" and "Agnus Dei". In the first four sections, all parts share
the same meter and tempo throughout. The 6-page "Agnus Dei" consists of
three subsections. In the middle subsection, all parts share a common meter
and tempo. There are simultaneous different tempi in the first and last
subsections of the "Agnus Dei".

In the first subsection of the "Agnus Dei", Choir I repeats a one-measure
10/8 phrase, four times. The first pass is taken at MM eighth note = 56.
During the course of the second and third passes, the tempo is increased
gradually from MM eighth note = 56 to MM eighth note = 72. The fourth pass
is taken at MM eighth note = 72. At the fifth eighth note of the Choir
I part, Choir III begins a one-measure 12/8 phrase that is repeated four
times. The first pass begins at MM eighth note = 70 and gradually decelerates
until reaching the tempo MM eighth note = 50 at the beginning of the fourth
pass. The fourth pass is taken at MM eighth note = 50. The following instruction
is given: "After conductor starts groups I + III in their respective tempi,
their future internal co-ordination should be assigned to a chosen member
of each group". Twelve seconds after Choir III begins (or roughly between
the 5th and 6th eighth note of Choir III), Choir II sings a 3-measure phrase
at MM eighth note = 60, and then, after a fermata of six to eight seconds,
it sings a 3-measure phrase that begins at MM sixteenth note = 100, accelerates
to MM sixteenth note = 112, and then ends with a rallentando to an unspecified
tempo. The subsection ends with all singers of Groups I and II holding
their last note. Group I is instructed to hold its last note through that
of Group III.

The third and final subsection of the "Agnus Dei" is sung by the sopranos
and altos. This subsection begins with the three altos holding their last
note from the previous subsection. Together, each soprano begins a one-measure
10/8 phrase that is repeated 3 times. The first pass is at MM eighth note
= 66. The second and third passes are marked "rall.ad lib." for Groups
I and III and "ad lib." for Group II. After cueing this entry, the conductor
cues the simultaneous entry of the three altos. Each has a different phrase.
Alto 1 sings his phrase "3 times: Molto rall. 2+3rd times", Alto 2 sings
his phrase "3 times: Poco rall. 2+3rd times", and Alto 3 sings his phrase
"Twice rall. 2nd time". When all sopranos have completed their repetitions,
they give a signal and then sing a final phrase. The start times of the
notes of the first repetition of the alto phrases and the final soprano
phrase occur in a specified order across all parts using a form of proportional
notation.

In the "Performing Notes", Ferneyhough writes:

In the Agnus Dei the three vocal groups work largely independently
of each other. In the passages involving multiple superimposed and conflicting
tempi, each group should assign the task of conductor to one of its own
members. No extra conductor is required for group two.

In the last [sub]section each individual singer pursues
his or her own course to the end, after repeating their material for the
given number of times. Any extra instructions are provided in context.

Ferneyhough, Brian 1969-71. Firecycle Beta: Symphonic Torso for Two
Pianos and Orchestra with Five Conductors. Ricordi.

This piece calls for a main conductor and four sub-conductors. The
players are partitioned into two groups (Orchestra 1 and 2) and each group
is partitioned into two sub-groups: a string ensemble and a mixed ensemble
of soloists. The main conductor (MC) is flanked by sub-conductors 1 (SC1)
and 2 (SC2) who are positioned in front of orchestra 1 and 2, respectively.
Sub-conductors 3 (SC3) and 4 (SC4) are positioned within orchestra 1 and
2, respectively. Multiple conductors are used in the following sections:
Processional(MC,
SC1, SC2), Sequence a (SC1, SC2, SC3, SC4),
Sequence b (MC,
SC3), Alleluia IIIb (all 5 conductors) and
Amen a (MC, SC3,
SC4). An introductory section titled "Special Instructions for the Conductors"
provides details regarding cueing and coordination. For example, the subsection
that pertains to sub-conductors 3 and 4 begins as follows:

These [SC3 and SC4] remain, in large part, dependant upon
[SC1] and [SC2] for their points of entry. The line of sight between each
pair should, therefore, be very clear [SC3 coordinates with SC1, SC4 with
SC2].

Sequence a, b. Although fully conventionally notated,
the materials (lower strings) are nevertheless dependant upon the speed
of movement of Sub-conductors 1 and 2 for their frequency of entry. Either
these latter give signals for each new entry, or else Sub-conductors 3
and 4 take their cues from the score itself at appropriate moments. As
soon as entry has been made, each conductor continues in his own, independently
prescribed tempo.

The section concludes with the following note:

In sections where several conductors are acting at one
and the same time, the vertical layout of the material does not necessarily
reflect the coincidence of sound-actions. Within the scope of the general
guidelines given above, the conductors move at their own individual speeds
through the material assigned to them.

Ghent, Emmanuel 1964. Triality: for Violin, Trumpet and Bassoon: in
Conjunction with a Polynome: 2 Studies.

The performance notes for this piece include the following instructions:
"It is intended that this music be played with the assistance of a polynome
... an instrument which can simultaneously beat up to 7 different tempi".
The score is available at the University
of Chicago Library (call number: fM362.G411T81) and the SUNY
at Buffalo Music Library (call numbers: Triality: M 5546 G341-3 t,
Triality II: M 5546 G341-3 t(2)).

Like Ghent's earlier work
Dithyrambos,
this piece employs pre-recorded audio beat cues which are distributed to
each player during a performance. The ensemble is partitioned into six
pairs of instruments. The instruments in a pair share a common tempo throughout
and the tempo of each pair is allowed to change independently of that of
any other pair. The audio cue signals enable the pairs to remain synchronized
with each other and with three channels of pre-recorded music. The six
initial metronome markings are in the ratio 5:3 (i.e. MM thirty-second
note = 555.6, MM sixteenth note = 333.3, MM sixteenth note = 200, MM eighth
note = 120, MM quarter note = 72, and MM half note = 43.2). The score is
available at The New
York Public Library for the Performing Arts (call number: JNG 79-227).

Ghent, Emmanuel 1969. Helices for Violin, Piano and Tape.

In some sections of this piece, two different and simultaneous metronome
markings are indicated, and at some points, accelerandos and decelerandos
are specified for one instrument while the other instrument proceeds at
a constant rate. Most of the notation conventions described in Ghent's
article Programmed Signals to Performers: A
New Compositional Resource are not used here. Instead, synchronization
is achieved by other means. For example, where the piano is to play at
a tempo of MM quarter note = about 64 while the violin plays at MM quarter
note = 116, the following instruction is given: "Piano: Play even sixteenths
at MM quarter note = about 64 attacca into the next measure. If too soon,
cut a few sixteenths; if late, repeat as required.". Later, when the violin
is to play at a tempo of MM quarter note = 96 while the piano plays septuplets
at MM quarter note = 109.7, the violin is given the instruction: "The tempo
and note placement are intended as guides and need not be slavishly followed.".
Also, at one point the page turner is instructed to use a stop-watch during
a 16-second duration of tape sound to cue the entry of the piano. And at
one point, a 25-second tape solo is notated against a timeline that runs
from 0 through 25. The score was published in Volume 8 of The
SCI Journal of Music Scores and is available at The
New York Public Library for the Performing Arts (call number: JMM 76-29
v. 8).

The piece consists of a sequence of 659 overlapping phrases, each with
an independent tempo selected from the metronome scale. The following tempi
are used: MM 4-note = 52, 58, 66, 76, 88, 100, 112, 126 and 144. Note:
In this score, a given note that starts at a given time-scale point i is
said to be an "n-note" (e.g. 1-note, 2-note, 3-note, etc.) if the note
ends at the time-scale point i+n.

No particular tempo predominates or serves as a reference. Each phrase
is notated on a separate page using a timetable. A zero-based time scale
of numbered points is given along the left and right edges of each timetable.
The time at which a given phrase is to occur, relative to other phrases,
is specified by an equation that indicates coincident time-scale points.
For example, for phrase 16, "1 = ¶15.3" indicates that time-scale
point 1 of phrase 16 occurs at the same time as time-scale point 3 of phrase
15. (Note: In this score, the symbol ¶ means "phrase".)

For most phrases the tempo is constant. For some phrases, the tempo
changes continuously as an accelerando or decelerando. Such transitions
are either geometric or sinusoidal. For example, the tempo of phrase 33
decreases geometrically from MM 4-note = 144 to MM 4-note = 76.

Immediately after phrase 1 ends, phrases 2 and 3 begin simultaneously
at a tempo of MM 4-note = 66 and MM 4-note = 52, respectively. While these
phrases are being played, phrase 5 begins at a tempo of MM 4-note = 112.
Immediately after phrase 8 ends, phrases 9 and 10 begin simultaneously
at a tempo of MM 4-note = 58 and MM 4-note = 112, respectively. The tempo
of phrase 9 increases geometrically to MM 4-note = 126, while phrase 10
continues at a constant tempo. Simultaneous different tempo markings of
this type occur throughout the piece.

On pages 71 through 80 of this work (marks 46 through 50), the solo
viola and other instruments are written in a meter of 3/4 at a tempo of
MM quarter note = 66. Simultaneously, the harp and harpsichord are written
in 4/4 at MM quarter note = 80. In this section of the score, the harp
and harpsichord parts have been reduced in size (both vertically and horizontally)
by an appropriate amount in order to scale the width of measures. Page
breaks have been placed at viola barlines and at various beat fractions
in the measures for the harp and harpsichord.

The strings which represent night sounds and darkness play Molto Adagio
throughout while the rest of the orchestra (which represents a variety
of other sounds that one might hear in the late 1800's in Central Park)
accelerates gradually from Molto Adagio to Allegro molto during measures
64 through 118. In a note in the score, Ives writes:

From measure 64 on, until the rest of the orchestra has
played measure 118, the relation of the string orchestra's measures to
those of the other instruments need not and cannot be written down exactly,
as the gradual accelerando of all but the strings cannot be played in precisely
the same tempi each time.

This is a programmatic piece in which a theme that represents "The
Perennial Question of Existence" is posed repeatedly by the trumpet and
the four flutes respond with material that represents the human hunt for
"The Invisible Answer". Throughout, a muted string ensemble represents
"The Silences of the Druids - Who Know, See and Hear Nothing". The trumpet
and strings are written in 4/4 at MM quarter note = 50 throughout. The
six flute answers, each of which is two to six bars long, are also written
in 4/4. Each successive answer is faster and the last accelerates, according
to the instructions: Adagio, Andante, Allegretto, Allegro, Allegro molto,
and Allegro - Accel. to Presto. All three barlines of the two-measure first
answer coincide with (and are connected through) the barlines of the trumpet.
In the same way, the leading barlines of the first two measures of the
four-measure second answer coincide with those of the trumpet. No barlines
of the fourth, fifth or sixth answers coincide with the trumpet. The following
temporal instructions are provided by Ives in the Foreward:

This part [the flute answers] need not be played in the
exact time position indicated. It is played in somewhat of an impromptu
way; if there is no conductor, one of the flute players may direct their
playing.

The flutes will end their part approximately near the
position indicated in the string score; but in any case, "The Last Question"
should not be played by the trumpet until "The Silences" of the strings
in the distance have been heard for a measure or two. The strings will
continue their last chord for two measures or so after the trumpet stops.
If the strings shall have reached their last chord before the trumpet plays
"The Last Question", they will hold it through and continue after, as suggested
above.

"The Answers" may be played somewhat sooner after each
"Question" than indicated in the score, but "The Question" should be played
no sooner for that reason.

Ives, Charles E. 1898-1912. Three Harvest
Home Chorales.

This work consists of three chorales. On pages 3 through 12 of the
second chorale, "Lord of the Harvest", the bass (and at times some other
instruments and voices) play at a speed that is 50% faster than that of
the other instruments and voices. All parts are written in 4/4 but those
which are faster are written as three consecutive half-note triplets that
span two 4/4 measures. Only one tempo term is specified at any given time,
but the duration of a given type of note is not the same in all parts at
all points in time. Ives uses arrows in the score to indicate the times
at which the half-note triplets are to begin in the faster part, relative
to half-note triplets in the slower part.

Ives, Charles E. 1909-16. Symphony No.
4.

The first page of the second movement contains a number of simultaneous
different tempi. (Note: The description given here refers to the version
of the score that was published in 1929 in New Music Edition 2(2).
This version differs from that published in 1965 by Associated Music Publishers.
In the later version, meters and barlines have been changed and two conductors
are required.) On the first page, one tempo term Allegretto is used along
with one metronome marking MM dotted-quarter note (of 6/8) = about 50.
Other tempi are indicated by the use of polymeters. For example, in the
time of five 6/8 measures, the primo orchestra piano has five 2/4 measures,
the bassoons have two 7/4 measures, and the strings have three 4/4 measures.
At the same time, an accelerando in the solo piano is indicated by a sequence
of seven 6/8 measures of decreasing width. In the "Conductor's Note", Ives
writes:

The recitative of the basses controls this page...The
solo piano from the third measure to end of page may not be played in the
exact time relation indicated by the measure divisions, but there is rather
a brief accelerando and crescendo and an easing down towards end of bass
recitative...

Later, in Section 8 of the second movement, some instruments continue
at Adagio while others begin playing Allegro and gradually accelerate through
the entire section. For this section, Ives writes:

The instruments are divided here into two separate orchestras;
the lower continuing the proceeding adagio, while the upper, including
woodwind, brass, tympani and both pianos, breaks suddenly in [at allegro]...Both
groups may keep in the time relation indicated on this page, but at about
the beginning of the next page the upper orchestra begins to play gradually
faster and faster until the "collapse" indicated on page 29, but which
will occur sooner - perhaps towards the end of page 28...During this passage
it may be advisable to have one of the players in the upper orchestra act
as a separate conductor.

Also, in the third and fourth measures of Section 8, the piccolo part
has a sequence of three quarter-note triplets that span two 4/4 measures.
As in Three Harvest Home Chorales,
the times at which these triplets begin is indicated by arrows, which in
this case, are directed from ordinary eighth-note triplets that begin on
a beat in 4/4 and are positioned above the piccolo staff.

For Section 20, Ives writes:

All the percussion with the saxophones or bassoons play
in Sec. 20 as a single and independent group. There may be a slight ritardando
as well as a decrescendo in this passage which may extend into Sec. 22.
If so, one of their number acting as leader for these few measures will
simplify the playing.

Also, in the "Conductor's Note", Ives suggests that the various rhythms
"would better stand out in their perspective" if the various players were
spatially separated.

In the fourth movement, there are polymetric textures for which the
parts do not share a common barline. The movement begins with a group of
percussionists playing "very slowly" in 4/4. Against this, the orchestra
plays in 6/2. Each 6/2 measure of the orchestra spans two 4/4 measures
of the percussion ensemble. Here, a whole note in the percussion ensemble
has the same duration as a dotted whole note in the orchestra, and thus,
these two ensembles play at a different tempo (though independent tempo
markings are not explicitly given in Ives' original manuscript).

The "Preface" of the full orchestral version edited by James B. Sinclair
(1976) contains the following notes on page vii which pertain to the second
section titled Putnam's Camp, Redding, Connecticut (Here, Sinclair
has added information (in brackets []) regarding the conductor Nicholas
Slonimsky who premiered the work in 1931.):

[This "Conductor's Note" was written in the early 1930's
by Ives with some collaboration by Slonimsky...]

[The following note is Slonimsky's except perhaps the
last paragraph. Slonimsky was particularly intrigued by the juxtaposition
of tempi here. He argued in numerous letters to Ives that the passage should
be engraved as an actual bi-tempo event, requiring the conductor to beat
conflicting patterns. Ives' refusals were politely firm until he allowed
Slonimsky his wish on the condition that an "ossia" line appear in the
score and parts showing how the passage can be performed in relation to
a single beat pattern. This edition restores Ives' original notation.]

At the measure after H, Page 35 [ii., m.68], the orchestra
is divided into two parts going on in two different rhythms until the last
two measures, Page 38 [ii., m.82-83]: the alla breve of the piano, drums,
bassoon, trumpet and second viola (or horn) is going 25% faster than the
rest of the orchestra, the result being that four beats of the former equals
three beats of the latter.

The conductor may beat the four-four time with his right
hand, and the alla breve with his left hand. The down-beat of the first
bar of the alla breve (left hand) falls in with the third beat of the 4/4
(right hand); the down-beat of the second bar of the alla breve (left hand)
falls in with the second beat of the right hand; the down-beat of
the third bar of the alla breve (left hand) falls in with the first
beat of the right hand, - in other words, here both down-beats come together.
In the following bar, the downbeat of the left hand falls with the fourth
beat of the right hand, then again with the third beat, etc. The piano,
drums, bassoon, trumpet and horn follow the conductor's left hand, the
rest - his right hand.

The up-beat of the left hand falls between the fourth
beat of the corresponding 4/4 bar and the first beat of the following bar,
and so forth, half of the alla breve bar being equal to 3/4 divided by
two, that is, 3/8. It is not difficult if the conductor will at first place
a mental accent on the third beat of 4/4, then on the second beat, and
so forth, as if following a waltz time while going through the motions
of conducting a quickstep.

Should the conductor prefer, he may beat only the main
line without bothering about the second rhythm part, a provision having
been made in the orchestral parts in case this is done.

Ives, Charles E. 1911-51. Universe Symphony.

Sketches of this unfinished work which are in the Collection of Ives
Manuscripts at the Irving
S. Gilmore Music Library of Yale
University Library, indicate that a large number of simultaneous different
meters were to be set against a common barline. It is not known whether
Ives would have notated a finished version of this with various tuplets
against a common meter and common tempo (as in Elliott Carter's String
Quartet No. 5), or with simultaneous different meters each with different
tempo markings (as in Elliott Carter's String
Quartet No. 1). Realizations of this work have been created by
Larry
Austin and Johnny Reinhard. (Note: The Reinhard realization which is
titled Charles Ives Universe Symphony Realized Exclusively on the Extant
Sketches by Johnny Reinhard (1994) has been excluded from this bibliography
because it is notated using a common tempo throughout.) An index to Ives
manuscripts that was compiled by John Kirkpatrick is available at Yale
University Library (call number: Ref. ML134 I95 K51+).

The score for this work consists of four pages of German text by Ligeti
along with translations in English (by Sid McLauchlan) and French (by Louise
Duchesneau). There are "Instructions for Performance" and "Notes". The
piece calls for 100 mechanical metronomes and two or more "interpreters".
The metronomes are divided evenly among the interpreters. Without the audience
present, the interpreters initialize the metronomes for performance. They
set the metronomes to different speeds between MM 50 and MM 144. They wind
each with four half-turns (from an unwound state) and place them on stage
on top of chairs, tables, pianos, or other "suitable resonators". Then,
they set all metronomes in motion "as simultaneously as possible", after
which the interpreters leave the stage and the audience enters the hall.
After 15 to 20 minutes, the piece ends when all metronomes have stopped
ticking.

Since all metronomes are wound by roughly the same amount, the faster
metronomes stop before slower ones. In his "Notes", Ligeti observes that
the piece may be thought of as consisting of the following three phases:
uniformity
resulting from the "collective blur of the individual periods" which is
heard as continuous sound, gradual structuring when complex rhythms
become perceptible after some metronomes have stopped, and uniformity
when the last metronome plays alone.

In measures 14 through 31 of the first movement of this four-movement
work, different tempo markings are indicated simultaneously. Here, a new
tempo is introduced with each successive measure. The conductor cues each
new tempo while some instrument is instructed to remain at the previous
tempo. The process continues until measure 21 at which time the following
tempi are in play (the tempi are listed by the order in which they are
introduced): MM quarter note = 92 (clarinet), 84 (bass clarinet), 76 (flute),
100 (piano), 60 (horn), 80 (organ), 66 (oboe d'amore) and 54 (trombone).
A footnote (p.34) states that each player should do their best to estimate
their proper tempo and "should not be synchronized by the conductor". Also,
here it is indicated that "The vertical relationships of the score are
of no relevance for these instruments (Fl., Clar., Clar. bass, Org., Pf.)...".
A similar pattern occurs in measures 35 through 39 of the third movement
where the tempi MM quarter note = 56, 60, 66, and 72 are used simultaneously.
Later in the same movement, simultaneous tempi accumulate in the same way
in measures 46 through 59. A footnote (p.80) for this passage states the
following:

The players left by the conductor on their own keep as
accurately as possible to their individual tempo (MM quarter note = 60
or MM quarter note = 80): however, since they cannot measure the tempo,
but only estimate it, tempo fluctuations may occur. Synchronization of
the (non-conducted) parts is not to be striven for; in fact, slight shifts
in the metre are welcome.

This work is in three sections. Throughout, the choir is divided into
two sub-choirs with the following composition: Section I: Choir I = Choir
II = {S1,S2;A;T1,T2;B}; Section II: Choir I = Choir II = {S1,S2;A1,A2;T1,T2;B1,B2};
Section III: Choir I = {S1,S2;A1,A2;T;B} and Choir II = {S+A,T+B}.

Throughout Section I, the tempo of Choir I is MM dotted-half note =
64 while that of Choir II is MM half note = 64. The meter for Choir I is
6/4 while that of Choir II is 2/2, and all parts share a common barline.
All measures are partitioned into 2 equal parts with a dashed barline.
At the beginning, the following tempo marking is given for each choir:
[dotted-half note] = [half note] = 64. Moderato meccanico.

In Section II, the two choirs share the same tempo marking throughout.

In Section III, five simultaneous tempi are used. The Bass of Choir
I begins at MM quarter note = 90 (Tempo giusto). On the third beat of measure
5 of the Bass, the Alto of Choir I begins at MM quarter note = 160 (Allegro
Vivace). At measure 11 of the Alto, the Soprano of Choir I begins at MM
quarter note = 140 (Allegro con anima). At measure 17 of the Alto, the
Tenor of Choir I begins at MM quarter note = 110 (Allegro moderato). At
measure 61 of the Alto, the Soprano and Alto of Choir II begin a unison
part at MM quarter note = 190 (Vivacissimo). The Tenor and Bass of Choir
II sing the same line as the Soprano and Alto through to the end, but lag
the Soprano and Alto by one quarter note. Arrows are used to indicate entry
times. All parts are in 4/4 and each part has an independent sequence of
measure numbers and every measure is numbered. Measures are frequently
fractured across system breaks. In each system, a left barline is drawn
through the beginning of all staves immediately after the opening clefs.
It is solid through groups of staves that begin on a barline and otherwise,
it is dashed. A similar matching barline is drawn at the end of the previous
system. Since the five tempi have a common divisor of 10, a duration consisting
of four measures of the Alto of Choir I (i.e. 16 beats) serves as a unit
of organization to which the tempi of the other parts may be coordinated.

In the (German) performance notes for Section III, Ligeti indicates
that each of the five groups (S (#1), A (#2), T (#3) and B (#4) of Choir
I and Choir II (#5)) is to have an assistant conductor which can be one
of the singers in the group if desired. The main conductor cues the entry
of each of the five groups, after which each assistant conductor maintains
the tempo for their group with the assistance of a metronome (a light metronome
is suggested). The main conductor may serve as the conductor of Group 2
(Alto of Choir I) while cueing the other groups along the way. Alternatively,
it is suggested that a 4-track tape player may be used to play pre-recorded
pulses for the assistant conductors of Groups 1 through 4 while the main
conductor leads Group 5 separately, or an 8-track tape player (or a digital
recorder) may be used to mechanically coordinate all five tempi. In either
case, the conductors would follow clicks heard in headphones or blinking
lights triggered by the pre-recorded pulses. Here, Ligeti indicates a preference
for lights.

This work consists of eleven sections. A common meter and tempo is
used throughout, except in the following sections (Note: The page numbers
given here are for the 1998 edition of the score published by Alphonse
Leduc, which is in two parts):

Section II. On pages 28 through 30, the following tempo terms are used
simultaneously for six flutes: "Presque vif" (Fl. 1), "Un peu vif" (Fl.
2), "Vif" (Fl. 3), "Modéré" (Fl. 4), "Un peu lent" (Fl. 5),
"Bien modéré" (Fl. 6). Each flute is associated with a different
bird: Orphean warbler (Fl. 1), spotted morning warbler (Fl. 2), chorister
robin (Fl. 3), Natal robin (Fl. 4), Rupell's robin-chat (Fl. 5), and musician
wren (Fl. 6). Throughout the passage, the tempo of each flute is constant,
except the fifth flute, which plays through the following sequence of tempi:
"Un peu lent", "Vif", "Un peu vif", "Un peu lent", "Un peu vif", "Un peu
lent", "Un peu vif", "Vif", "Un peu vif", "Très modéré".
The flutes enter in numerical order, staggered. Their entries are cued
by the conductor. In each part, a bold arrow is directed up or down to
the note or rest that is to be cued. A similar passage occurs on pages
42 through 45. Here again, the six flutes play at different tempi, and
the tempo of the fifth flute changes throughout while that of the other
flutes remains constant. In these sections, no parts are metered. Common
music notation is spaced to indicate the time at which notes are to occur.

Section VIII. On pages 23 through 29, flute 1 plays an unmeasured solo
passage at "(Vif) hors tempo - au signe du chef" while the rest of the
orchestra plays measured material at "Bien modéré (MM eighth
note = 100)". A similar structure occurs on pages 45 through 59. However,
here flute 2 begins a short time after flute 1 at "(Presque vif) hors tempo
- au signe du chef". In these passages, the conductor cues the entries
of the flutes. Here, the flutes are associated with garden warblers. At
other times in this section, there are passages in which one instrument
plays in a different tempo while other instruments hold a note throughout
(pp. 37, 62 and 65).

Section IX. This section (pp. 103-126) is polytempo throughout. It is
scored for 18 woodwinds (3 piccolos, 6 flutes, a solo flute, 2 piccolo
clarinets in Eb, 6 clarinets in Bb), a triangle, and a small cymbal. Each
woodwind instrument is associated with one or more of 25 different species
of bird. Each species is represented by some characteristic thematic material
of a given tempo, which is indicated by a tempo term. The entry of each
bird-song phrase is cued by the conductor. Different bird songs overlap
in time. A wide variety and large number of different tempi occur simultaneously.
For example, at the end of page 10, the following tempi are used simultaneously:
"Vif" (Picc. 2, Fl. 3, Fl. 4); "Modéré, un peu vif" (Fl.
1, Fl. 2); "Modéré" (Fl. 5, Picc. Cl. 2, Cl. 2, Cl. 4); "Un
peu vif" (Fl. 6, Cl. 5); "Un peu lent" (Solo Flute); "Presque vif" (Cl.
3). Although some simultaneous phrases have the same tempo, those which
do are not synchronized. That is, their beats do not occur at the same
time. Throughout this section, parts are unmeasured, and the time at which
notes are to occur is indicated spatially.

Musgrave, Thea 1966. Chamber Concerto No. 2. London: Chester.

In various unmeasured sections of this piece, simultaneous different
tempi are specified. For example, on page 51, the clarinet has a tempo
marking of "sempre MM half note = 84 (faster than violin)" where the prevailing
tempo marking for the violin is "Moderato MM half note = 60". On page 54,
the clarinet and violin play at MM half note = 100 while the cello plays
at "MM half note = about 76 (slower than Cl./Vl.)" and the piano is at
"MM quarter note = 132 Tempo di presto misterioso". In this section, the
piano part is measured with dashed barlines while the other parts are unmeasured.

Nancarrow, Conlon. Study No. 2 for Player Piano. Santa Fe: Soundings.

The following tempo markings are used (Note: Here, the tempi are listed
in the order in which they first appear and meters are given in parentheses):
MM quarter note = 69 (3/4), MM eighth note = 115 (5/8), MM quarter note
= 86-1/4 (2/4), MM dotted-eighth note = 115 (6/16), MM five sixteenth notes
= 69 (10/16), and MM dotted-quarter note = 57-1/2 (6/8). The tempo numbers
are in the ratio 10:12:15:20. During one interval of time, five different
tempo markings are in effect simultaneously. These accumulate in the following
order: MM quarter note = 69, MM eighth note = 115, MM quarter note = 86-1/4,
MM five sixteenth notes = 69, and MM dotted-quarter note = 57-1/2. Each
tempo occurs in a constant meter throughout with the exception of a 5/16
measure where the tempo of one voice changes from MM dotted-eighth note
= 115 to MM five sixteenth notes = 69. Common barlines occur frequently.

Nancarrow, Conlon. Study No. 6 for Player Piano. Santa Fe: Soundings.

A single tempo marking of "measure = 60" is given which indicates that
there are to be 60 measures per minute. Each part is measured with a different
meter that remains constant throughout, and all parts share a common barline
throughout. Thus, the duration of a given type of note is different in
each part. The simultaneous meters are 4/8, 5/8 and 3/4. The implied simultaneous
tempi are: MM eighth note = 240 (= 4*60), MM eighth note = 300 (= 5*60),
and MM quarter note = 180 (= 3*60).

Nancarrow, Conlon. Study No. 9 for Player Piano. Santa Fe: Soundings.

There are up to three simultaneous independent parts. The parts have
independent metrical schemes and various combinations of two or three tempi
are used simultaneously, in the following order:

To indicate that an unmeasured pause is to occur after a given phrase,
an upward-sloping line is drawn through the middle three staff lines after
the last barline of the phrase. This notational device is used in many
of Nancarrow's scores.

There are two simultaneous parts with independent tempi and meter.
The tempo of one part is MM quarter note = 88 while that of the other part
is MM quarter note = 110. The ratio between the two tempi is 4:5. The faster
part begins after, and ends before, the slower part. Each part has two
voices or staves. The staves of the faster part are positioned above those
of the slower part. Very few barlines are common to both parts. System
breaks are placed at a barline of the slower part more often than at a
barline of the faster part.

This is a two-voice canon. The voices begin and end together, and both
sound throughout the piece. The material for each voice is played twice,
once at one tempo and then again at a second tempo. For one voice, the
first pass is at MM half note = 165 and the second pass is at MM half note
= 220. For the other voice, the first pass is at MM half note = 220 and
the second pass is at MM half note = 165. The ratio between the tempo numbers
is 3:4. Changes in meter that occur in one voice do not coincide with those
of the other voice. Common barlines are drawn wherever the barlines of
the voices coincide exactly.

This piece consists of a sequence of three sections. The tempo of the
first section is MM half note = 84. For the second section, the tempo is
MM half note = 140. In the third section, the first two sections are played
simultaneously. The ratio between the two tempi is 3:5. In each section,
there are four voices. Each voice has a constant meter that is different
from that of the other voices and equal to one of the following: 4/2, 5/2,
6/2 and 7/2.

In this work, there are three independent voices. All voices sound
throughout the piece. The following tempi occur simultaneously: MM half
note = 138, 172-1/2 and 230. The ratio between these tempi is 12:15:20.
Sometimes, two of the voices share the same tempo, but at no time during
the piece are all three voices at the same tempo. The three voices begin
and end together (i.e. the first and last notes occur at the same time
in all voices). The piece begins with the tempi assigned as follows: MM
half note = 172-1/2 (voice 1), 138 (voice 2) and 230 (voice 3). During
the course of the piece, tempo changes are made in the following order:
the tempo of the second voice is changed to MM half note = 230; the tempo
of the third voice is changed to MM half note = 138; the tempo of the second
voice is changed to MM half note = 172-1/2; and finally, the tempo of the
first voice is changed to MM half note = 230. Thus, the tempo of the second
voice is swapped with that of the third voice, and then swapped with that
of the first voice. Changes in meter that occur in one voice do not coincide
with those of the other voices. Common barlines are drawn wherever the
barlines of the voices coincide exactly, which is rarely the case.

This is a two-voice canon. The tempo of one voice is MM half note =
168 while that of the other voice is MM half note = 224. The ratio between
these tempi is 3:4. The slower voice begins alone and then is joined by
the faster voice. The two voices end together (i.e. the last note of the
slower voice occurs at the same time as that of the faster voice). Changes
in meter that occur in one voice do not coincide with those of the other
voice. Common barlines are drawn wherever the barlines of the voices coincide
exactly, which is frequently the case.

This a canon for three voices each with its own tempo. The ratio between
the three tempi is 12:15:20 and the tempi are as follows: MM quarter note
= 144 (voice number 1), MM quarter note = 180 (voice number 2), and MM
quarter note = 240 (voice number 3). The voices enter, one at a time, from
slowest to fastest. The entry times of the faster voices, each of which
is a transposed version of the first voice, are delayed so that all voices
end the piece at almost the same time. The first voice is unmeasured until
the second voice enters, after which all parts are measured, though time
signatures are not indicated explicitly. Common barlines occur across all
parts before and after blocks (or hypermeasures) consisting of the simultaneous
combination of twelve eighth notes at MM quarter note = 144 (voice 1),
fifteen eighth notes at MM quarter note = 180 (voice 2), and twenty eighth
notes at MM quarter note = 240 (voice 3). Also, common barlines are drawn
through voices 1 and 2 at every four eighth notes of voice 1 and five eighth
notes of voice 2. And there are common barlines through voices 2 and 3
at every three eighth notes of voice 2 and four eighth notes of voice 3.

In some sections, the following tempi occur simultaneously: MM dotted-quarter
note = 149-1/3, 160 and 170-2/3. In other sections, the following tempi
are used together: MM dotted-quarter note = 224, 240 and 256. In either
case, the ratios between the tempi are 14:15:16. All parts are metered,
and at many points in time, all parts have the same meter. Common barlines
occur infrequently.

One voice has a constant tempo while each of the other voices accelerates
or decelerates at one of four possible rates that are indicated by tempo
markings such as "accel. - 6%" or "rit. - 11%". Different rates of acceleration
and deceleration occur simultaneously. Tempo markings are positioned above
the staff with which they are associated. The accelerations and decelerations
are geometric. That is, in a given passage for which a gradual change in
tempo is to occur, the ratio between successive durations is a constant
that is determined by the indicated percentage. Proportional notion is
used throughout for all parts. Barlines are used in some sections, and
where they are used, they are positioned independently in each part and
no time signature is indicated.

This is a three-voice canon where each voice has a different tempo.
The ratio between the three tempi is 21:24:25 and the tempi are as follows:
MM quarter note = 105 (voice number 1), MM quarter note = 120 (voice number
2), and MM quarter note = 125 (voice number 3). The voices enter, one at
a time, from slowest to fastest, and the entry times of the faster voices
are such that all voices end the piece together. As is generally the case
in Nancarrow's studies for player piano, the staves are ordered in each
system by tempo, with the topmost staff being the fastest. All voices are
measured with a meter of 2/4 throughout. Common barlines occur across all
parts before and after blocks (or hypermeasures) consisting of the simultaneous
combination of 21 measures at MM quarter note = 105 (voice 1), 24 measures
at MM quarter note = 120 (voice 2), and 25 measures at MM quarter note
= 125 (voice 3). Also, common barlines are drawn through voices 1 and 2
at every seven measures of voice 1 and eight measures of voice 2.

This piece has two simultaneous parts with one or two voices per part.
The tempo of one part is MM quarter note = 140*sqrt(2) and that of the
other part is MM quarter note = 280. Here, sqrt(2) means the square root
of 2. The ratio between the tempi is sqrt(2):2. Both parts are measured,
and meters are changed frequently and independently in each part. Since
the tempo ratio is irrational, there are few barlines that are common to
both parts. Common barlines occur at the left barline of the first measure,
and before a measure that is near the middle of the piece during a phrase
in one part that, after an unmeasured pause, begins at a time that is not
a beat or (rational) sub-beat of the other part.

At almost all times during this work, there are simultaneous different
tempi. At times, as many as five different tempi occur at once. For example,
at the beginning of the piece, five tempi accumulate, one at a time, in
the following order: MM quarter note = 119, 170, 204, 283-1/3 and 340.
During the course of the piece, the following tempi are used: MM quarter
note = 79-1/3, 85, 102, 113-1/3, 119, 141-2/3, 170, 204, 238, 283-1/3 and
340. The ratios between the tempi may be seen more clearly by dividing
the tempo numbers by 5-2/3 to yield the quotients: 14, 15, 18, 20, 21,
25, 30, 36, 42, 50 and 60. Parts are metered independently and common barlines
are drawn whenever the barlines of parts coincide exactly. One section
of the piece is written with proportional notation.

This is a four-voice canon where each voice has a different tempo.
The ratio between the three tempi is 17:18:19:20 and the tempi are as follows:
MM whole note = 85 (voice number 1), MM whole note = 90 (voice number 2),
MM whole note = 95 (voice number 3), and MM whole note = 100 (voice number
4). The voices enter, one at a time, from slowest to fastest, and the entry
times of the faster voices are such that all voices end the piece together.
All system breaks occur at a barline in the slowest part which is the lowest
staff in each system. All voices are measured throughout with various meters
(e.g. 4/4, 2/4 and 6/4). Common barlines occur across all parts before
and after blocks (or hypermeasures) consisting of the simultaneous combination
of 17 measures at MM whole note = 85 (voice 1), 18 measures at MM whole
note = 90 (voice 2), 19 measures at MM whole note = 95 (voice 3), and 20
measures at MM whole note = 100 (voice 4).

The following tempi are used: MM quarter note = 150, 160-5/7, 168-3/4,
180, 187-1/2, 200, 210, 225, 240, 250, 262-1/2 and 281-1/4. At several
points in the piece, all twelve tempi occur simultaneously. For example,
the piece begins with twelve voices sounding, one for each tempo. The ratios
between the tempo numbers and lowest number 150 are as follows: 1/1, 15/14,
9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4 and 15/8. This series of ratios
is identical to that which is associated with a tempo scale proposed by
Henry Cowell in his book New
Musical Resources. Cowell's scale is based upon ratios that occur
in an overtone series.

Throughout the piece, phrases are metered independently in each voice
and common barlines are rare. The piece consists of a sequence of 12 sections,
though they are not labeled as such. In some sections, the entry times
of successive phrases are staggered. In such cases, the first beat of the
first measure, or at least the first beat of some measure, of a (child)
phrase that is to enter coincides with a beat or sub-beat of a (parent)
phrase in another voice. For example, on page 14, the first beat of the
second measure of the child phrase at tempo MM quarter note = 180 coincides
with the first beat of a measure of the parent phrase at tempo MM quarter
note = 168-3/4. Such times of coincidence are indicated with a common barline
or a dotted vertical line that runs from the parent phrase to the child
phrase. In some sections, all phrases are aligned at the first barline
of the section (e.g. Section 1: p.1-4, or Section 2: p.4-8) or at the last
barline of the section (e.g. Section 3: p.8-11).

This work calls for two non-synchronized pianos. The piece consists
of two sections labeled A and B. There are no tempo markings. Instead,
tempo is specified indirectly in terms of the duration of each section.

Section A is a two-voice canon with a duration of 4 minutes. The ratio
between the tempi of the voices is e:pi (or e/pi which may be pronounced
"e over pi") where "e" is the base of the natural system of logarithms
(approx. 2.71828) and "pi" is the ratio of the circumference to the diameter
of a circle (approx. 3.14159). In his book The
Music of Conlon Nancarrow, Kyle Gann notes that this tempo ratio
may be approximated by 13:15 (Note: This approximation is slightly better
than 19:22 which would be obtained by dividing a common rational approximation
for e (19/7) by a common rational approximation for pi (22/7)). The entry
of the faster voice is delayed so that the two voices end simultaneously.
Both voices are metered throughout and meters are changed frequently (at
almost every bar). Except for the left barline of the final bar, there
are no common barlines.

In Section B, Section A is performed again while simultaneously, on
a second player piano, Section A is performed more slowly. The slow version
of Section A is to have a duration that is 20 seconds longer than the original.
Thus, the ratio between the tempi in the slow version and the original
is 12:13. In Section B, the entry of the original version of Section A
occurs after that of the slow version.

Raxach, Enrique 1969, 1989. Inside Outside. Donemus.

On page 2, the following tempi occur simultaneously: MM quarter note
= 76 in 2/4 (cello 1), MM quarter note = 60 in 5/16 (cello 2), MM quarter
note = 60 in 3/4 (cello 3), MM dotted-quarter note = 60 in 6/8 (cello 4),
MM quarter note = 63 in 3/4 (double bass 3), MM dotted-quarter note = 60
in 3/8 (double bass 4), MM quarter note = 48 in 2/4 (double bass 5), and
MM eighth note = 104 in 2/8 (double bass 6). These instruments begin together
and play in this way for about 17 seconds while the violins and some violas
and cellos play sustained notes (some with quarter-tone vibrato) at the
prevailing tempo. Also, during this passage double basses 1 and 2 play
different unmetered phrases that contain accelerandos and decelerandos
(indicated with feathered beams) which are at times simultaneous and opposing.
This page is quoted in the book Modern
Rhythmic Notation by Gardner Read (p. 170).

The piece begins with the first piano repeatedly playing a sequence
of twelve sixteenth notes on the pitches: E / F# / B / C# / D \ F# \ E
/ C# \ B \ F# / D \ C#. Note: Here "X / Y" ("X \ Y") indicates that Y is
a higher (lower) pitch than X by an octave or less. The odd-numbered notes
of the sequence are played with the left hand while the even-numbered notes
are played with the right. Gradually, the second piano fades in, in unison
with piano 1. Then, the second piano is instructed to "accel very slightly"
until the following two notes are occurring at the same time: the first
note of the phrase played by piano 1 (E), and the second note of the phrase
played by piano 2 (F#). When the players become aligned in this way, the
second piano is instructed to "hold tempo 1". During the times at which
it is accelerating, the part of the second piano is notated with dotted
staff lines only. Next, piano 2 accelerates again (henceforth, indicated
with the abbreviation "a.v.s.") until its phrase is two sixteenths ahead
of piano 1. Again, at this time piano 2 is instructed to hold tempo 1.
This process is repeated twelve times until the two pianos are in unison
again. Then the second piano fades out and rests.

After some number of repetitions of the twelve-note phrase, piano 1
begins repeating an eight-note phrase that is a sub-phrase of the twelve-note
phrase, consisting of the 1st through 6th notes followed by the 9th and
12th notes (E / F# / B / C# / D \ F# / B / C#). After some repetitions,
piano 2 fades in and plays the following eight-note phrase against piano
1: E / E \ A / B / D / E \ A / B. Then, piano 2 performs eight acceleration
transformations as before until the two pianos are aligned again as they
were before the transformations. Then piano 1 fades out while piano 2 continues
to repeat its eight-note phrase.

Piano 2 then plays the first six notes of its eight-note phrase, once
(E / E \ A / B / D / E), after which it plays the last four notes of this,
repeatedly (A / B / D / E). Piano 1 fades in, in unison with Piano 2. Piano
2 performs four acceleration transformations as before until the two pianos
are in unison again. Then after some repetitions, both players end together.

Throughout the piece, the number of repetitions for each bar is specified
approximately as a range. For example, "x12-18" is used to indicate that
a given bar is to be repeated about 12 to 18 times. Players are instructed
to proceed to the next bar after having done the following: "listen to
the two voice relationship and as you hear it clearly and have absorbed
it, move on to the next bar".

A single metronome marking of MM quarter note = about 144 is given.
During several intervals of time, a given instrument is instructed to accelerate
very slightly while the other instruments remain in tempo. Instruments
that accelerate do so until the one-measure looped phrase that is being
played is advanced in time by one eighth note. An instruction such as "(x6-16)"
is given to indicate that an acceleration is to occur over the duration
of 6 to 16 repetitions of the looped phrase. In sections where a given
instrument is to accelerate, the 5-line staff for the part is replaced
by a 5-line, 2-dimensional pattern of dots.

This piece contains one metronome marking of MM quarter note = about
176. When an organ is not at rest, it is playing one of a small set of
one-measure (4/4) patterns repeatedly. This piece begins with the first
and second organs playing the same pattern in unison. At various points
in the piece, the second organ is instructed to "accel. very slightly".
In the "Directions for Performance", it is specified that in these sections
the tempo of the second organ is to be increased gradually "as slowly as
possible" until it is playing one eighth note earlier than before, while
all other instruments continue at the original tempo. After the eighth-note
out-of-phase relationship is established between the first and second organ,
the second organ returns to the original tempo. After eight transformations
of this type, the second organ is once again playing in unison with the
first organ and the piece ends.

Reich, Steve 1971. Drumming Part Two for Three Marimbas and Female Voices.

In this work, there are nine marimba players positioned around three
marimbas. At various points during the piece, one marimba player is instructed
to slightly increase their tempo so as to gradually move ahead of the other
players by a full quarter note. At other times, players that have become
out of phase in this way, are instructed to slightly increase or decrease
their tempo in order to slowly move back into unison with other players.

In various sections of this work, simultaneous different tempo markings
are specified. For example, this occurs on the following pages in the 77-page
full score (Universal Edition 12697LW): pp. 10-12, 22-24, 37-42 and 44-53.

The tempo of some passages is indicated with a metronome marking while
that of others is indicated with a German textual phrase such as: "so schnell
wie möglich" (as fast as possible), "so langsam wie möglich"
(as slow as possible), "schnell - verlangsamen" (fast - slowing down),
"langsam - beschleunigen" (slow - quickening). Regarding these various
tempo markings, Stockhausen writes the following in the "Performing Directions"
(Note: Here, the term "group" means passage or phrase):

1. Tempi in the form of metronome-markings are to be observed
exactly.

2. The tempo "as fast as possible" applies to the shortest
note-value of a group. Within this group the chosen mensural unit remains
constant, while the effective speed varies with the note-values employed.

3. The tempo "as slow as possible" is determined by the
breathing-capacity of the woodwind-player, who has to perform a particular
series of groups in one breath. The greatest total duration for the series
determines the constant mensural value.

4. When the tempo is "fast - slowing down", the mensural
units (mainly [sixteenth note]) at the beginning and end of the group are
to stand roughly in the proportion 4:1 (e.g. [sixteenth note] ca. 240 -->
[sixteenth note] ca. 60). Thus the length of the group determines the rate
of slowing-down. The smallest note-value in the group decides how quickly
the beginning can be played. While the mensural units become steadily longer,
the effective speed can, because of the varying note-values, alter irregularly.

5. When the tempo is "slow - quickening", the shortest
possible mensural unit at the end of the group is first to be settled:
from this is determined the mensural unit, four times as long, for the
beginning of the group (e.g. [sixteenth note] ca. 70 <-- [sixteenth
note] ca. 280).

As an approximate control measure the groups with relative
tempi (2.-5.) are drawn in the score to proportionate scales.

At some points, simultaneous different metronome markings are specified.
For example, this occurs on page 44 where the following markings are used:
MM eighth note = 64 (oboe), MM eighth note = 80 (English horn), MM eighth
note = 112 (clarinet). The same markings are used simultaneously in one
measure on page 46 for the English horn, bassoon and clarinet, respectively.
Here, the three parts share common barlines with meters 4/16, 5/16, and
7/16, and they occur simultaneously with an unmeasured part for oboe that
has the marking "schnell - verlangsamen" (fast - slowing down). Later,
on pages 52-3 the following metronome markings are used simultaneously:
MM eighth note = 64 (oboe and flute), 80 (English horn), 96 (clarinet),
and 112 (bassoon).

Dashed vertical lines are used to indicate common barlines or beats.
Such lines are drawn through a staff at the point at which a common barline
or beat occurs.

This piece requires three orchestras, each with its own conductor.
In general, some note or sound within the part of one of the sounding orchestras
serves as a cue for a conductor to begin beating a silent measure to establish
a new tempo and meter, and prepare his orchestra for entry.

A tempo scale is used that consists of the tempi MM quarter note = 60,
63.5, 67, 71, 75.5, 80, 85, 90, 95, 101, 107, 113.5 and 120. Analogous
to pitch scales, this is approximately an "equal-tempered" tempo scale
from 60 through 120 with 12 steps where the ratio between any two successive
tempi in the scale is roughly equal to the 12th root of 2. (More precisely,
the tempi in an equal-tempered scale of this type would be: 60.00, 63.57,
67.35, 71.35, 75.60, 80.09, 84.85, 89.90, 95.24, 100.91, 106.91, 113.26
and 120.00.) In this piece, almost all tempi are derived from this scale
or a multiple of 2 or 1/2 thereof. One exception occurs at mark 54 on page
48 in orchestra 1. Here the tempo MM eighth note = 157.5 is used in order
to align the barlines of a 7/8 measure of orchestra 1 with that of a 4/4
measure at MM quarter note = 90 in orchestra 2.

Approximations in timing have been made throughout the score to simplify
the notation. For example, on page 33, the right barline of a measure of
4 beats at MM quarter note = 90 in orchestra 3 is made to coincide with
that of a measure with 5 beats at MM quarter note = 113.5 in orchestra
2. Actually, these two durations are not equal, but treating them as such
greatly simplifies the placement of system breaks as measures are not fractured
across systems. Another example occurs on page 47 where the right barline
of a 7/8 measure at MM quarter note = 80 in orchestra 1 is made to coincide
with that of a 4/4 measure at MM quarter note = 90 in orchestra 2. In this
case, the approximation is clarified to be in the tempo of orchestra 1
as this is written as MM quarter note = 80 (78.75).

The six percussionists are positioned at the corners of a hexagon around
the audience. In the English "Directions" section, Xenakis writes: "p.
14 m. 191: The 6 metronomes must be exactly synchronized (or else electronically
controlled).". During various sections of the piece, the tempi of the players
are independent. For example, in measures 193 to 215, tempo changes are
gradually and independently introduced into each part. The tempi that occur
simultaneously are MM quarter note = 38, 40, 42, 58, 74 and 78. From measures
220 to 227, these tempi are used again, but here they begin simultaneously
at measure 220 after a 6-second silence which is presumably used to allow
each player to receive a preparatory cue to establish their individual
tempo. In measures 250 and 296-297, the following tempi are juxtaposed:
MM quarter note = 44, 52, 56, 60, 64 and 68.

Articles

Austin, Larry 1985. “Charles Ives's Life Pulse Prelude for Percussion Orchestra:
A Realization for Modern Performance from Sketches for his Universe Symphony.”
Percussive
Notes: Research Edition 23(6): 58-84.

Austin describes the process by which his Life
Pulse Prelude was created based upon sketches by Ives. Copies of
relevant sketch pages are included and their significance discussed. Also,
the method by which synchronization was accomplished in live performance
is described (p.78) as well as other methods used in earlier recorded versions
that were created for tape performance (p.73). In this piece, different
simultaneous tempi occur because all parts share a common right barline
but not a common meter. The number of beats per measure which occur simultaneously
are: 1, 2, 4, 8, 3, 6, 9, 12, 5, 10, 7, 14, 11, 22, 13, 17, 19, 23, 29
and 31. In performance, 12 pre-recorded tape tracks of independent beat-cue
patterns were used to generate cues for the following tempi: MM quarter
note = 60, 75, 82.5, 90, 105 and 97.5; MM half note = 67.5; MM eighth note
= 127.5, 142.5, 172.5, 217.5 and 232.5. The click pitch and the duration
between clicks was varied to distinguish the first and last beats of a
measure from intermediate beats. For example, the sequence of click pitches
used for the 31 beats of a 31/8 measure was |C---c---c---c---c---c---c---ceg|
where 'C' is the pitch that is an octave above the pitch 'c', 'e' and 'g'
are in the octave between 'c' and 'C', and '-' represents a beat for which
no click was sounded. Similarly, the sequence used for 13/4 measures was
|Ccccccccccceg|. In performance, it was found that open air headphones
which were used to deliver cue signals to performers, were unacceptable
during extremely loud passages.

Cowell, Henry 1927. “The Impasse of Modern Music.”
The Century Magazine
114(6): 671-7.

Cowell discusses various forms of innovation that characterize the
Western art music of the early 20th century. Regarding the use of simultaneous
different tempi, he writes:

In orchestral music we may very practically evolve combinations
in musical tempo which cannot well be written for any single instrument.
As a matter of musical feeling a slow note in allegro is very different
from a fast note in adagio, and music which is to convey more than a simple
emotion, can be written far more effectively by using different tempos
in its different parts. The effort of conveying different emotions in the
same tempo is a tremendous handicap to a composer, and is largely responsible
for the stilted and artificial quality of most opera. Take, for example,
the quartette from Rigoletto, where Verdi attempts to portray the emotions
of four different characters. It would be far more realistic if each part
were written in the tempo called for by the emotion of the character. The
contralto part should be sung in allegro, the tenor in allegretto, the
baritone in adagio and the soprano in andante. Instead, all are sung in
the same tempo, the conflict of emotions is reduced to an average and the
standardized tempo reduces, rather than advances, the dramatic action.

Ghent, Emmanuel 1967. “Programmed Signals to
Performers: A New Compositional Resource.” Perspectives of New Music
6(1): 96-106. Reprinted in Boretz, Benjamin and Edward T. Cone (Eds.) 1976.
Perspectives
on Notation and Performance. New York: Norton.

This article describes a system for transmitting audio cue signals
to musicians to coordinate the performance of rhythmically complex music.
An associated notation system is proposed whereby conventional music notation
is set against and scaled to a timeline that runs throughout the score
above each system. A rhythmic cue line is provided above the staff for
each part to indicate beat groupings and the times at which various cues
are to be transmitted. Cue signals are used for the following purposes:
to indicate downbeats and upbeats (with signals for downbeats being longer
than those for upbeats), to establish a new tempo before notes are to be
played (as a few preparatory beats), to signal the entire rhythmic pattern
of a complicated phrase before it is to be played (e.g. for klangfarbenmelodie
that has an aperiodic rhythm), and to provide a reassuring checkpoint cue
or an alerting cue just before the end of a long rest (in the form of a
rapid succession of three cue signals). Initially, cue signals were generated
during performance by a device called the
coordinome.
In later performances, signals were played back and decoded from a single
track of magnetic tape on which cue signals for all instruments had been
recorded, with the signal for each instrument being at a different frequency.

This article describes the coordinome which is a device that was designed
and developed at New York University
by Ghent and Stein G. Raustein. The device was originally intended to be
used to coordinate performers of rhythmically complex music, or to synchronize
performers with tape music. This article discusses these uses along with
additional applications in the areas of rhythmic training in education,
the control of theatrical elements (e.g. lighting), and the real-time control
of sound generating and processing equipment.

This brief article proposes a system consisting of centrally-coordinated
wearable computer displays on which musical scores and beat indicators
could be shown to facilitate the performance of rhythmically complex pieces
such as polytempo music. In this system, beat-cue information which is
supplied to each performer would be derived from, but not necessarily equal
to, the beat established by a human conductor equipped with a wired baton
(e.g. one of the available Baton
Interfaces such as the Mathews/Boie
Radio Baton or the Buchla
Lightning) or a lead performer equipped with a suitable controller
such as a wired foot pedal. The article lists the musical implications
and potential advantages of such a system. Note: This article is contained
in the web site Finale:
IWBNI (It Would Be Nice If) which is a set of 724 suggestions for improving
the music notation program Finale by Coda
Music, a division of Net4Music.
Thus, in this context the system is presented as a potential feature for
the music notation software program Finale.

The derivation, spelling, usage and meaning of the word polytempo is
given. Related words such as multi-tempo, polytempic, polychronic and polytemporal,
which have been used in place of polytempo by various authors are also
discussed. Non-English spellings of the word polytempo are also given.

This is an overview of the various approaches that have been taken
by composers to facilitate the performance of polytempo music. The following
subjects are discussed: loose coordination, notational solutions, technology-assisted
conducting, and electronic realizations.

This article proposes some suggestions as to how technology might be
used to coordinate the performance of polytempo music. The following subjects
are discussed: semi-automatic conducting, content-rich audio cues, visual
cues from artificial conductors, tactile cues, multi-sensory cues, and
wireless connections.

This article is a survey of the different forms of tempo scales that
have been used in polytempo music. A tempo scale is defined as a
collection of tempi arranged in order from lowest to highest or from highest
to lowest. The following types of scales are discussed: tempo-term scales,
the metronome scale, harmonic scales, equal tempered scales, and augmentation/diminution
scales.

In the subsection "Sound of No Restraint", a MIDI-based click-track
system is described that enables an ensemble to remain synchronized while
each player performs at a different tempo. First, a MIDI file is created
that contains one track for each player. A unique MIDI channel is assigned
to each track. An appropriate sequence of click notes is inserted into
each track. During performance, a sequencer program plays the MIDI file.
The MIDI output of the sequencer is routed to the MIDI input of a sampler
that has multiple monophonic audio outputs. The sampler is configured so
that a unique MIDI channel is assigned to each audio output. The audio
outputs are routed through small headphone amplifiers to open air headphones.
A cowbell sample is used for click sounds. In the concluding section "Some
Present and Future Experiments", Ligeti proposes a system by which a set
of drum pads could be used to conduct an ensemble. In this case, the audio
output derived from each pad would be routed to a different player.

Stockhausen, Karlheinz 1959. "How Time Passes." die Reihe 3 (Musical
Craftsmanship): 10-40. Bryn Mawr: T. Presser. English translation of the
original German version by Cornelius Cardew.

In this article, Stockhausen puts forward an equal-tempered, eight-octave
duration scale that is a low-frequency extension of an equal-tempered pitch
scale for which the frequency of the lowest pitch is 32 Hz. To represent
this duration scale, he uses the following metronome markings MM whole
note = 60, 63.6, 67.4, 71.4, 75.6, 80.1, 84.9, 89.9, 95.2, 100.9, 106.9,
113.3 and 120. These tempi are used in combination with the following note
symbols: sixteenth note, eighth note, quarter note, half note, whole note,
double whole-note, quadruple whole-note, and octal whole-note. In this
tempo scale, there are twelve tempi per octave and the ratio between any
two successive tempi is equal to the twelfth root of two. A tempo scale
that is closely related to this is used in Stockhausen's Gruppen
for 3 Orchestras.

In discussing realizations of works based upon this scale of durations,
Stockhausen writes (p.25):

...the realization of our example requires three chordal
instruments, or, better, three orchestras, that would at times play independently
of each other, at differing tempi, orientating themselves to the others
only at the points where they entered. Before each entry, each group -
or each conductor - could prepare for the next tempo without difficulty
(with a metronome). Spatial separation would result naturally from the
need to make various time-strata appreciable.

On page 32, he considers the notion of four simultaneous passages with
different tempo structures: a passage with a constant tempo, a passage
that is to be played as fast as possible, a passage during which there
is an accelerando, and a passage during which there is a decelerando.

Sullivan, Timothy 1997. “Multiple Tempi: A Survey and Method.” Conference
of the Canadian University Music Society.

This article discusses a method by which simultaneous different tempi
may be notated so that all parts are written against at common reference
tempo, and in each measure, all parts have the same meter. The method consists
of a thorough application of the techniques of augmentation and diminution,
and uses notational devices similar to those found in scores by Elliott
Carter. In Figure 14, an extensive table is given to indicate the note
duration that would be required to notate one beat, or various fractions
of a beat (i.e. 1/5, 1/4, 1/3, 1/2, 2/3 and 3/4), of a tempo that stands
in a given ratio to the reference tempo, given that the duration of a beat
in the reference tempo equals that of a quarter note. A wide range of possible
tempo ratios are listed in this table. Excerpts are quoted from the literature
in which different tempo markings are indicated simultaneously, including:
Karlheinz Stockhausen's Gruppen (Figure 7: five measures around
mark 28), Charles Ives' Putnam's Camp from Three Places in New
England (Figure 11: mark H to I), György Ligeti's Chamber Concerto
for 13 Instrumentalists (Figure 19: measures 13-23), and Harry Somers'
Those
Silent, Awe Filled Spaces (Figure 21: pages 16-21), and the proposed
method of notation is used to renotate these excerpts (except
Gruppen).
With this method, the duration of a given type of note is the same in all
parts, at each point in time. Sullivan has used these techniques in his
works Three Etudes in Multiple Tempi and Terrains I for Orchestra
(1999).

Willey uses the melody of Frère Jaques to illustrate various
tempo ratios between two voices. The concepts of consonant tempo ratios
(e.g. 2:1), dissonant tempo ratios (e.g. 32:31), convergence (i.e. voices
end synchronized), and divergence (i.e. voices begin synchronized) are
described. The relative degree of dissonance of a given tempo ratio is
expressed (loosely) in terms of the number of beats that occur between
consecutive coincident beats.

A detailed description of the structure of Nancarrow's Study No.
37 is given in a table that includes the following entries for each
section of the work: page number, duration, [entry] order of tempi, type
of canon (i.e. convergent, divergent, or "internal synch"), order of ranges
[or entry order of voices], interval of entrance, ambitus of voice material,
dynamic level, duration of entry notes, melodic character, texture, perceptibility
of canon, and contrast with previous section. Three excerpts are given
from this study (specifically, page 1 and two portions of Section V), all
of which depict instances of simultaneous different tempo markings.

The use of contrasting timbres and spatial positions for different voices
is explored in order to make the underlying canonic forms of Study No.
37 more apparent. Consideration is given to the relative degree to
which tempo and various note properties (i.e. start time, pitch, duration,
dynamic level, timbre and spatial position) are a determining factor for
one to perceive a given passage as melodic (i.e. horizontal) vs. harmonic
(i.e. vertical). For this, concepts of Max Wertheimer's Gestalt psychology
are applied (e.g. proximity, similarity, direction [continuity] and common
destiny [common fate]).

Books, Theses and Dissertations

The section "Accelerando and rallentando" of the chapter titled "Time"
(p. 73) contains a discussion of various methods for notating a gradual
change of tempo in one part while other parts continue in strict tempo.
A "tempo glissando" in Elliott Carter's String Quartet No. 2 is
quoted.

Cowell, Henry 1930. New Musical
Resources. New York: Alfred A. Knopf.

In Section 6 "Tempo" and Section 7 "Scales of Rhythm" of Part II "Rhythm",
Cowell discusses the notion of combining different tempi simultaneously.
Also, some matters pertaining to situations in which the tempo of one part
remains constant while that of another simultaneous part is varied continuously,
are considered. The musical excerpts given in the figures for Examples
24 (p.93) and 26 (p.97) contain simultaneous different metronome markings.
It is suggested that the tempi of such polytempo textures be drawn from
a tempo scale that is constructed using ratios that are ordinarily associated
with the overtone series.

Epstein, David 1995. Shaping Time: Music, the Brain, and Performance.
New York: Schirmir.

This is an extensive scientific and artistic examination of various
topics related to time and tempo in music. It includes numerous case studies
of works from the Classical and Romantic Periods. The following excerpt
from "Chapter Eleven: Rubato" (p. 373) discusses simultaneous different
tempi in romantic rubato (Here, "romantic rubato" refers to a rubato style
whereby all elements (melody, harmony, rhythm) of a passage are affected
simultaneously):

Further implicit in both views of romantic rubato is the
fascinating phenomenon of two separate and different time controls operating
simultaneously as the phrase or other musical segment progresses. There
is the strict metric control of the beat - the hypothetically invariant
time unit that serves as the ground element for timing. Moving in parallel
with this is the pulse of various segments of the phrase, a pulse distorted
by rubato as a means of enhancing musical expression.

These two time controls, really systems of time
control, rapidly become dissynchronous and thus in conflict, thereby adding
excitement to the performance, for we feel ourselves pulled by each system,
internally stressed or contorted by the tensions that each exerts. A large
part of the gratification in good rubato playing lies in the eventual reconciliation
of these two systems, their return to phase synchrony.

Gann, Kyle 1995. The Music of Conlon
Nancarrow. Cambridge UK: Cambridge University Press.

In the chapter titled "Polymetric Notation" (pp. 168-173), the following
examples of pieces that contain simultaneous different tempi are discussed:
Lejaren Hiller's String Quartet No. 5, Hans Werner Henze's Compases
para preguntas ensimismadas for solo viola and orchestra, Girolamo
Arrigo's Fluxus, Marius Constant's Winds and Enrique Raxach's
Inside
Outside. Single-page excerpts of the Arrigo and Raxach works are quoted.
Rhythmic schematics are given for small portions of the Hiller and Constant
pieces.

Saunders, James E. 1996. “The Final Frontier: The Development of Polytempo
in the Music of Conlon Nancarrow.” Masters dissertation, Royal Northern
College of Music.

Abstract excerpt:
The development of Conlon Nancarrow's use of polytempo
is traced from the simple micro-pulse relationships in his early music
through to the more elaborate irrational number relationships of the later
pieces. The effects of his various approaches on the resulting music are
examined.

Stauffer, George B. 1997. The Mass in
B Minor: The Great Catholic Mass. New York: Schirmer; London: Prentice
Hall.

Instances in which the part scores for Bach's Mass in B Minor
contain simultaneous different tempi are discussed (p. 234 and p. 295 n.59).
A related article by Bernard D. Sherman titled Bach's Notation of Tempo
and Early-Music Performance: some reconsiderations is available at
http://www.kdsi.net/~sherman/bachtempo.htm.

Stone, Kurt 1980. Music Notation in the Twentieth
Century: A Practical Guidebook. New York; London: Norton.

The notation of music with different simultaneous speeds is discussed
in Chapter III: "Duration and Rhythm: Preliminary Survey" and Chapter IV:
"Duration and Rhythm: Individual Items". In the section "Spatial Notation
versus Traditional (Symbolic) Notation" (p. 137), a figure is given to
show how spatial notation may be used to notate a simultaneous ritardando
and accelerando. In the section "Different Rates of Speed" (pp. 119-120),
the approximations used in Elliott Carter's String Quartet No. 2
are discussed (see also: p. 115). The multi-rate nature of this work is
described in greater detail in the section "Polymetric Combinations: Notation
by Reference Meter" (pp. 90-3) where two excerpts of the score are quoted.

This 317-page dissertation explores the rhythmic and textural techniques
of Conlon Nancarrow's Studies for Player Piano in general, and his
temporal dissonance in particular. It includes the following sections:
I. Introduction; II. Temporal Stratification in the Works of Other Composers;
III. Multidimensionality and Textural Strategies; IV. Temporal Dissonance;
V. Study No. 41; VI. Concluding Remarks.

Works Under Consideration

This is a list of works that are known or suspected to be suitable for
inclusion in the bibliography of polytempo music given here. They are being
researched.

Austin, Larry 1983-4. Sonata Concertante.

Austin, Larry. Charles Ives' Universe
Symphony as realized and completed by Larry Austin (1974-93). Also,
related works: Life Pulse Prelude (1984) and Life Pulse Prelude
for Live and Recorded Percussionists (1984/96).

Brant, Henry. Any work by Brant is a candidate for the bibliography given
here, especially spatial pieces and works completed after the initial version
of Antiphony One (1953), with the exception of the following pieces
which have been reviewed and judged to be unqualified for inclusion in
this bibliography: The Marx Brothers: 3 Faithful Portraits, Capriccio
by J. S. Bach, All Souls Carnival, Concerto for Alto Saxophone
Solo or Trumpet Solo with Nine Instruments, Stresses, Whoopee
in D Major, The $1,000,000 Confessions, American Commencement,
American
Requiem, Galaxy 2, Requiem in Summer, Millennium 1,
Signs
and Alarms, Partita for Flute and Piano, From Bach's Menagerie,
Four
Short Nature Pieces, and Atlantis.

June 10, 2001: Added the article “Beta Foly: Experiments with Tradition
and Technology in West Africa” by Lukas Ligeti and more works
under consideration. Excluded mensuration canons, augmentation and
diminution.